[<< wikibooks] Partial Differential Equations/Distributions
== Distributions and tempered distributions ==

Proof:
Let 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
   be a tempered distribution, and let 
  
    
      
        O
        ⊆
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle O\subseteq \mathbb {R} ^{d}}
   be open.
1.
We show that 
  
    
      
        
          
            T
          
        
        (
        φ
        )
      
    
    {\displaystyle {\mathcal {T}}(\varphi )}
   has a well-defined value for 
  
    
      
        φ
        ∈
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle \varphi \in {\mathcal {D}}(O)}
  .
Due to theorem 3.9, every bump function is a Schwartz function, which is why the expression

  
    
      
        
          
            T
          
        
        (
        φ
        )
      
    
    {\displaystyle {\mathcal {T}}(\varphi )}
  makes sense for every 
  
    
      
        φ
        ∈
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle \varphi \in {\mathcal {D}}(O)}
  .
2.
We show that the restriction is linear.
Let 
  
    
      
        a
        ,
        b
        ∈
        
          R
        
      
    
    {\displaystyle a,b\in \mathbb {R} }
   and 
  
    
      
        φ
        ,
        ϑ
        ∈
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle \varphi ,\vartheta \in {\mathcal {D}}(O)}
  . Since due to theorem 3.9 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   and 
  
    
      
        ϑ
      
    
    {\displaystyle \vartheta }
   are Schwartz functions as well, we have

  
    
      
        ∀
        a
        ,
        b
        ∈
        
          R
        
        ,
        φ
        ,
        ϑ
        ∈
        
          
            D
          
        
        (
        O
        )
        :
        
          
            T
          
        
        (
        a
        φ
        +
        b
        ϑ
        )
        =
        a
        
          
            T
          
        
        (
        φ
        )
        +
        b
        
          
            T
          
        
        (
        ϑ
        )
      
    
    {\displaystyle \forall a,b\in \mathbb {R} ,\varphi ,\vartheta \in {\mathcal {D}}(O):{\mathcal {T}}(a\varphi +b\vartheta )=a{\mathcal {T}}(\varphi )+b{\mathcal {T}}(\vartheta )}
  due to the linearity of 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
   for all Schwartz functions. Thus 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
   is also linear for bump functions.
3.
We show that the restriction of 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
   to 
  
    
      
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}(O)}
   is sequentially continuous. Let 
  
    
      
        
          φ
          
            l
          
        
        →
        φ
      
    
    {\displaystyle \varphi _{l}\to \varphi }
   in the notion of convergence of bump functions. Due to theorem 3.11, 
  
    
      
        
          φ
          
            l
          
        
        →
        φ
      
    
    {\displaystyle \varphi _{l}\to \varphi }
   in the notion of convergence of Schwartz functions. Since 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
   as a tempered distribution is sequentially continuous, 
  
    
      
        
          
            T
          
        
        (
        
          φ
          
            l
          
        
        )
        →
        
          
            T
          
        
        (
        φ
        )
      
    
    {\displaystyle {\mathcal {T}}(\varphi _{l})\to {\mathcal {T}}(\varphi )}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== The convolution ==

The convolution of two functions may not always exist, but there are sufficient conditions for it to exist:
Theorem 4.5:
Let 
  
    
      
        p
        ,
        q
        ∈
        [
        1
        ,
        ∞
        ]
      
    
    {\displaystyle p,q\in [1,\infty ]}
   such that 
  
    
      
        
          
            1
            p
          
        
        +
        
          
            1
            q
          
        
        =
        1
      
    
    {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}
   and let 
  
    
      
        f
        ∈
        
          L
          
            p
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle f\in L^{p}(\mathbb {R} ^{d})}
   and 
  
    
      
        g
        ∈
        
          L
          
            q
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle g\in L^{q}(\mathbb {R} ^{d})}
  . Then for all 
  
    
      
        y
        ∈
        O
      
    
    {\displaystyle y\in O}
  , the integral

  
    
      
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        f
        (
        x
        )
        g
        (
        y
        −
        x
        )
        d
        x
      
    
    {\displaystyle \int _{\mathbb {R} ^{d}}f(x)g(y-x)dx}
  has a well-defined real value.
Proof:
Due to Hölder's inequality,

  
    
      
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        
          |
        
        f
        (
        x
        )
        g
        (
        y
        −
        x
        )
        
          |
        
        d
        x
        ≤
        
          
            (
            
              
                ∫
                
                  
                    
                      R
                    
                    
                      d
                    
                  
                
              
              
                |
              
              f
              (
              x
              )
              
                
                  |
                
                
                  p
                
              
              d
              x
            
            )
          
          
            1
            
              /
            
            p
          
        
        
          
            (
            
              
                ∫
                
                  
                    
                      R
                    
                    
                      d
                    
                  
                
              
              
                |
              
              g
              (
              y
              −
              x
              )
              
                
                  |
                
                
                  q
                
              
              d
              x
            
            )
          
          
            1
            
              /
            
            q
          
        
        <
        ∞
      
    
    {\displaystyle \int _{\mathbb {R} ^{d}}|f(x)g(y-x)|dx\leq \left(\int _{\mathbb {R} ^{d}}|f(x)|^{p}dx\right)^{1/p}\left(\int _{\mathbb {R} ^{d}}|g(y-x)|^{q}dx\right)^{1/q}<\infty }
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  We shall now prove that the convolution is commutative, i. e. 
  
    
      
        f
        ∗
        g
        =
        g
        ∗
        f
      
    
    {\displaystyle f*g=g*f}
  .

Proof:
We apply multi-dimensional integration by substitution using the diffeomorphism 
  
    
      
        x
        ↦
        y
        −
        x
      
    
    {\displaystyle x\mapsto y-x}
   to obtain

  
    
      
        (
        f
        ∗
        g
        )
        (
        y
        )
        =
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        f
        (
        x
        )
        g
        (
        y
        −
        x
        )
        d
        x
        =
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        f
        (
        y
        −
        x
        )
        g
        (
        x
        )
        d
        x
        =
        (
        g
        ∗
        f
        )
        (
        y
        )
      
    
    {\displaystyle (f*g)(y)=\int _{\mathbb {R} ^{d}}f(x)g(y-x)dx=\int _{\mathbb {R} ^{d}}f(y-x)g(x)dx=(g*f)(y)}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Proof:
Let 
  
    
      
        α
        ∈
        
          
            N
          
          
            0
          
          
            d
          
        
      
    
    {\displaystyle \alpha \in \mathbb {N} _{0}^{d}}
   be arbitrary. Then, since for all 
  
    
      
        y
        ∈
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle y\in \mathbb {R} ^{d}}
  

  
    
      
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        
          |
        
        f
        (
        x
        )
        
          ∂
          
            α
          
        
        
          η
          
            δ
          
        
        (
        y
        −
        x
        )
        
          |
        
        d
        x
        ≤
        ‖
        
          ∂
          
            α
          
        
        
          η
          
            δ
          
        
        
          ‖
          
            ∞
          
        
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        d
        x
      
    
    {\displaystyle \int _{\mathbb {R} ^{d}}|f(x)\partial _{\alpha }\eta _{\delta }(y-x)|dx\leq \|\partial _{\alpha }\eta _{\delta }\|_{\infty }\int _{\mathbb {R} ^{d}}|f(x)|dx}
  and further

  
    
      
        
          |
        
        f
        (
        x
        )
        
          ∂
          
            α
          
        
        
          η
          
            δ
          
        
        (
        y
        −
        x
        )
        
          |
        
        ≤
        
          |
        
        f
        (
        x
        )
        
          |
        
      
    
    {\displaystyle |f(x)\partial _{\alpha }\eta _{\delta }(y-x)|\leq |f(x)|}
  ,Leibniz' integral rule (theorem 2.2) is applicable, and by repeated application of Leibniz' integral rule we obtain

  
    
      
        
          ∂
          
            α
          
        
        f
        ∗
        
          η
          
            δ
          
        
        =
        f
        ∗
        
          ∂
          
            α
          
        
        
          η
          
            δ
          
        
      
    
    {\displaystyle \partial _{\alpha }f*\eta _{\delta }=f*\partial _{\alpha }\eta _{\delta }}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Regular distributions ==
In this section, we shortly study a class of distributions which we call regular distributions. In particular, we will see that for certain kinds of functions there exist corresponding distributions.

Two questions related to this definition could be asked: Given a function 
  
    
      
        f
        :
        
          
            R
          
          
            d
          
        
        →
        
          R
        
      
    
    {\displaystyle f:\mathbb {R} ^{d}\to \mathbb {R} }
  , is 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
        :
        
          
            D
          
        
        (
        O
        )
        →
        
          R
        
      
    
    {\displaystyle {\mathcal {T}}_{f}:{\mathcal {D}}(O)\to \mathbb {R} }
   for 
  
    
      
        O
        ⊆
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle O\subseteq \mathbb {R} ^{d}}
   open given by

  
    
      
        
          
            
              T
            
          
          
            f
          
        
        (
        φ
        )
        :=
        
          ∫
          
            O
          
        
        f
        (
        x
        )
        φ
        (
        x
        )
        d
        x
      
    
    {\displaystyle {\mathcal {T}}_{f}(\varphi ):=\int _{O}f(x)\varphi (x)dx}
  well-defined and a distribution? Or is 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
        :
        
          
            S
          
        
        (
        
          
            R
          
          
            d
          
        
        )
        →
        
          R
        
      
    
    {\displaystyle {\mathcal {T}}_{f}:{\mathcal {S}}(\mathbb {R} ^{d})\to \mathbb {R} }
   given by

  
    
      
        
          
            
              T
            
          
          
            f
          
        
        (
        ϕ
        )
        :=
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        f
        (
        x
        )
        ϕ
        (
        x
        )
        d
        x
      
    
    {\displaystyle {\mathcal {T}}_{f}(\phi ):=\int _{\mathbb {R} ^{d}}f(x)\phi (x)dx}
  well-defined and a tempered distribution? In general, the answer to these two questions is no, but both questions can be answered with yes if the respective function 
  
    
      
        f
      
    
    {\displaystyle f}
   has the respectively right properties, as the following two theorems show. But before we state the first theorem, we have to define what local integrability means, because in the case of bump functions, local integrability will be exactly the property which 
  
    
      
        f
      
    
    {\displaystyle f}
   needs in order to define a corresponding regular distribution:

Now we are ready to give some sufficient conditions on 
  
    
      
        f
      
    
    {\displaystyle f}
   to define a corresponding regular distribution or regular tempered distribution by the way of

  
    
      
        
          
            
              T
            
          
          
            f
          
        
        :
        
          
            D
          
        
        (
        O
        )
        →
        
          R
        
        ,
        
          
            
              T
            
          
          
            f
          
        
        (
        φ
        )
        :=
        
          ∫
          
            O
          
        
        f
        (
        x
        )
        φ
        (
        x
        )
        d
        x
      
    
    {\displaystyle {\mathcal {T}}_{f}:{\mathcal {D}}(O)\to \mathbb {R} ,{\mathcal {T}}_{f}(\varphi ):=\int _{O}f(x)\varphi (x)dx}
  or

  
    
      
        
          
            
              T
            
          
          
            f
          
        
        :
        
          
            S
          
        
        (
        
          
            R
          
          
            d
          
        
        )
        →
        
          R
        
        ,
        
          
            
              T
            
          
          
            f
          
        
        (
        ϕ
        )
        :=
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        f
        (
        x
        )
        ϕ
        (
        x
        )
        d
        x
      
    
    {\displaystyle {\mathcal {T}}_{f}:{\mathcal {S}}(\mathbb {R} ^{d})\to \mathbb {R} ,{\mathcal {T}}_{f}(\phi ):=\int _{\mathbb {R} ^{d}}f(x)\phi (x)dx}
  :
Proof:
1.
We show that if 
  
    
      
        f
        ∈
        
          L
          
            loc
          
          
            1
          
        
        (
        O
        )
      
    
    {\displaystyle f\in L_{\text{loc}}^{1}(O)}
  , then 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
        :
        
          
            D
          
        
        (
        O
        )
        →
        
          R
        
      
    
    {\displaystyle {\mathcal {T}}_{f}:{\mathcal {D}}(O)\to \mathbb {R} }
   is a distribution.
Well-definedness follows from the triangle inequality of the integral and the monotony of the integral:

  
    
      
        
          
            
              
                
                  |
                  
                    
                      ∫
                      
                        U
                      
                    
                    φ
                    (
                    x
                    )
                    f
                    (
                    x
                    )
                    d
                    x
                  
                  |
                
                ≤
                
                  ∫
                  
                    U
                  
                
                
                  |
                
                φ
                (
                x
                )
                f
                (
                x
                )
                
                  |
                
                d
                x
                =
                
                  ∫
                  
                    
                      supp 
                    
                    φ
                  
                
                
                  |
                
                φ
                (
                x
                )
                f
                (
                x
                )
                
                  |
                
                d
                x
              
            
            
              
                ≤
                
                  ∫
                  
                    
                      supp 
                    
                    φ
                  
                
                ‖
                φ
                
                  ‖
                  
                    ∞
                  
                
                
                  |
                
                f
                (
                x
                )
                
                  |
                
                d
                x
                =
                ‖
                φ
                
                  ‖
                  
                    ∞
                  
                
                
                  ∫
                  
                    
                      supp 
                    
                    φ
                  
                
                
                  |
                
                f
                (
                x
                )
                
                  |
                
                d
                x
                <
                ∞
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\left|\int _{U}\varphi (x)f(x)dx\right|\leq \int _{U}|\varphi (x)f(x)|dx=\int _{{\text{supp }}\varphi }|\varphi (x)f(x)|dx\\\leq \int _{{\text{supp }}\varphi }\|\varphi \|_{\infty }|f(x)|dx=\|\varphi \|_{\infty }\int _{{\text{supp }}\varphi }|f(x)|dx<\infty \end{aligned}}}
  In order to have an absolute value strictly less than infinity, the first integral must have a well-defined value in the first place. Therefore, 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
      
    
    {\displaystyle {\mathcal {T}}_{f}}
   really maps to 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
   and well-definedness is proven.
Continuity follows similarly due to

  
    
      
        
          |
        
        
          T
          
            f
          
        
        
          φ
          
            l
          
        
        −
        
          T
          
            f
          
        
        φ
        
          |
        
        =
        
          |
          
            
              ∫
              
                K
              
            
            (
            
              φ
              
                l
              
            
            −
            φ
            )
            (
            x
            )
            f
            (
            x
            )
            d
            x
          
          |
        
        ≤
        ‖
        
          φ
          
            l
          
        
        −
        φ
        
          ‖
          
            ∞
          
        
        
          
            
              
                
                  ∫
                  
                    K
                  
                
                
                  |
                
                f
                (
                x
                )
                
                  |
                
                d
                x
              
              ⏟
            
          
          
            
              independent of 
            
            l
          
        
        →
        0
        ,
        l
        →
        ∞
      
    
    {\displaystyle |T_{f}\varphi _{l}-T_{f}\varphi |=\left|\int _{K}(\varphi _{l}-\varphi )(x)f(x)dx\right|\leq \|\varphi _{l}-\varphi \|_{\infty }\underbrace {\int _{K}|f(x)|dx} _{{\text{independent of }}l}\to 0,l\to \infty }
  , where 
  
    
      
        K
      
    
    {\displaystyle K}
   is the compact set in which all the supports of 
  
    
      
        
          φ
          
            l
          
        
        ,
        l
        ∈
        
          N
        
      
    
    {\displaystyle \varphi _{l},l\in \mathbb {N} }
   and 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   are contained (remember: The existence of a compact set such that all the supports of 
  
    
      
        
          φ
          
            l
          
        
        ,
        l
        ∈
        
          N
        
      
    
    {\displaystyle \varphi _{l},l\in \mathbb {N} }
   are contained in it is a part of the definition of convergence in 
  
    
      
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}(O)}
  , see the last chapter. As in the proof of theorem 3.11, we also conclude that the support of 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   is also contained in 
  
    
      
        K
      
    
    {\displaystyle K}
  ).
Linearity follows due to the linearity of the integral.
2.
We show that 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
      
    
    {\displaystyle {\mathcal {T}}_{f}}
   is a distribution, then 
  
    
      
        f
        ∈
        
          L
          
            loc
          
          
            1
          
        
        (
        O
        )
      
    
    {\displaystyle f\in L_{\text{loc}}^{1}(O)}
   (in fact, we even show that if 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
        (
        φ
        )
      
    
    {\displaystyle {\mathcal {T}}_{f}(\varphi )}
   has a well-defined real value for every 
  
    
      
        φ
        ∈
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle \varphi \in {\mathcal {D}}(O)}
  , then 
  
    
      
        f
        ∈
        
          L
          
            loc
          
          
            1
          
        
        (
        O
        )
      
    
    {\displaystyle f\in L_{\text{loc}}^{1}(O)}
  . Therefore, by part 1 of this proof, which showed that if 
  
    
      
        f
        ∈
        
          L
          
            loc
          
          
            1
          
        
        (
        O
        )
      
    
    {\displaystyle f\in L_{\text{loc}}^{1}(O)}
   it follows that 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
      
    
    {\displaystyle {\mathcal {T}}_{f}}
   is a distribution in 
  
    
      
        
          
            
              D
            
          
          
            ∗
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}^{*}(O)}
  , we have that if 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
        (
        φ
        )
      
    
    {\displaystyle {\mathcal {T}}_{f}(\varphi )}
   is a well-defined real number for every 
  
    
      
        φ
        ∈
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle \varphi \in {\mathcal {D}}(O)}
  , 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
      
    
    {\displaystyle {\mathcal {T}}_{f}}
   is a distribution in 
  
    
      
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}(O)}
  .
Let 
  
    
      
        K
        ⊂
        U
      
    
    {\displaystyle K\subset U}
   be an arbitrary compact set. We define

  
    
      
        μ
        :
        K
        →
        
          R
        
        ,
        μ
        (
        ξ
        )
        :=
        
          inf
          
            x
            ∈
            
              
                R
              
              
                d
              
            
            ∖
            O
          
        
        ‖
        ξ
        −
        x
        ‖
      
    
    {\displaystyle \mu :K\to \mathbb {R} ,\mu (\xi ):=\inf _{x\in \mathbb {R} ^{d}\setminus O}\|\xi -x\|}
  
  
    
      
        μ
      
    
    {\displaystyle \mu }
   is continuous, even Lipschitz continuous with Lipschitz constant 
  
    
      
        1
      
    
    {\displaystyle 1}
  : Let 
  
    
      
        ξ
        ,
        ι
        ∈
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle \xi ,\iota \in \mathbb {R} ^{d}}
  . Due to the triangle inequality, both

  
    
      
        ∀
        (
        x
        ,
        y
        )
        ∈
        
          
            R
          
          
            2
          
        
        :
        ‖
        ξ
        −
        x
        ‖
        ≤
        ‖
        ξ
        −
        ι
        ‖
        +
        ‖
        ι
        −
        y
        ‖
        +
        ‖
        y
        −
        x
        ‖
         
         
         
         
         
        (
        ∗
        )
      
    
    {\displaystyle \forall (x,y)\in \mathbb {R} ^{2}:\|\xi -x\|\leq \|\xi -\iota \|+\|\iota -y\|+\|y-x\|~~~~~(*)}
  and

  
    
      
        ∀
        (
        x
        ,
        y
        )
        ∈
        
          
            R
          
          
            2
          
        
        :
        ‖
        ι
        −
        y
        ‖
        ≤
        ‖
        ι
        −
        ξ
        ‖
        +
        ‖
        ξ
        −
        x
        ‖
        +
        ‖
        x
        −
        y
        ‖
         
         
         
         
         
        (
        ∗
        ∗
        )
      
    
    {\displaystyle \forall (x,y)\in \mathbb {R} ^{2}:\|\iota -y\|\leq \|\iota -\xi \|+\|\xi -x\|+\|x-y\|~~~~~(**)}
  , which can be seen by applying the triangle inequality twice.
We choose sequences 
  
    
      
        (
        
          x
          
            l
          
        
        
          )
          
            l
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (x_{l})_{l\in \mathbb {N} }}
   and 
  
    
      
        (
        
          y
          
            m
          
        
        
          )
          
            m
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (y_{m})_{m\in \mathbb {N} }}
   in 
  
    
      
        
          
            R
          
          
            d
          
        
        ∖
        O
      
    
    {\displaystyle \mathbb {R} ^{d}\setminus O}
   such that 
  
    
      
        
          lim
          
            l
            →
            ∞
          
        
        ‖
        ξ
        −
        
          x
          
            l
          
        
        ‖
        =
        μ
        (
        ξ
        )
      
    
    {\displaystyle \lim _{l\to \infty }\|\xi -x_{l}\|=\mu (\xi )}
   and 
  
    
      
        
          lim
          
            m
            →
            ∞
          
        
        ‖
        ι
        −
        
          y
          
            m
          
        
        ‖
        =
        μ
        (
        ι
        )
      
    
    {\displaystyle \lim _{m\to \infty }\|\iota -y_{m}\|=\mu (\iota )}
   and consider two cases. First, we consider what happens if 
  
    
      
        μ
        (
        ξ
        )
        ≥
        μ
        (
        ι
        )
      
    
    {\displaystyle \mu (\xi )\geq \mu (\iota )}
  . Then we have

  
    
      
        
          
            
              
                
                  |
                
                μ
                (
                ξ
                )
                −
                μ
                (
                ι
                )
                
                  |
                
              
              
                
                =
                μ
                (
                ξ
                )
                −
                μ
                (
                ι
                )
              
              
            
            
              
              
                
                =
                
                  inf
                  
                    x
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                ‖
                ξ
                −
                x
                ‖
                −
                
                  inf
                  
                    y
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                ‖
                ι
                −
                y
                ‖
              
              
            
            
              
              
                
                =
                
                  inf
                  
                    x
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                ‖
                ξ
                −
                x
                ‖
                −
                
                  lim
                  
                    m
                    →
                    ∞
                  
                
                ‖
                ι
                −
                
                  y
                  
                    m
                  
                
                ‖
              
              
            
            
              
              
                
                =
                
                  lim
                  
                    m
                    →
                    ∞
                  
                
                
                  inf
                  
                    x
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                
                  (
                  
                    ‖
                    ξ
                    −
                    x
                    ‖
                    −
                    ‖
                    ι
                    −
                    
                      y
                      
                        m
                      
                    
                    ‖
                  
                  )
                
              
              
            
            
              
              
                
                ≤
                
                  lim
                  
                    m
                    →
                    ∞
                  
                
                
                  inf
                  
                    x
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                
                  (
                  
                    ‖
                    ξ
                    −
                    ι
                    ‖
                    +
                    ‖
                    x
                    −
                    
                      y
                      
                        m
                      
                    
                    ‖
                  
                  )
                
              
              
                (
                ∗
                )
                
                   with 
                
                y
                =
                
                  y
                  
                    m
                  
                
              
            
            
              
              
                
                =
                ‖
                ξ
                −
                ι
                ‖
              
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}|\mu (\xi )-\mu (\iota )|&=\mu (\xi )-\mu (\iota )&\\&=\inf _{x\in \mathbb {R} ^{d}\setminus O}\|\xi -x\|-\inf _{y\in \mathbb {R} ^{d}\setminus O}\|\iota -y\|&\\&=\inf _{x\in \mathbb {R} ^{d}\setminus O}\|\xi -x\|-\lim _{m\to \infty }\|\iota -y_{m}\|&\\&=\lim _{m\to \infty }\inf _{x\in \mathbb {R} ^{d}\setminus O}\left(\|\xi -x\|-\|\iota -y_{m}\|\right)&\\&\leq \lim _{m\to \infty }\inf _{x\in \mathbb {R} ^{d}\setminus O}\left(\|\xi -\iota \|+\|x-y_{m}\|\right)&(*){\text{ with }}y=y_{m}\\&=\|\xi -\iota \|&\end{aligned}}}
  .Second, we consider what happens if 
  
    
      
        μ
        (
        ξ
        )
        ≤
        μ
        (
        ι
        )
      
    
    {\displaystyle \mu (\xi )\leq \mu (\iota )}
  :

  
    
      
        
          
            
              
                
                  |
                
                μ
                (
                ξ
                )
                −
                μ
                (
                ι
                )
                
                  |
                
              
              
                
                =
                μ
                (
                ι
                )
                −
                μ
                (
                ξ
                )
              
              
            
            
              
              
                
                =
                
                  inf
                  
                    y
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                ‖
                ι
                −
                y
                ‖
                −
                
                  inf
                  
                    x
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                ‖
                ξ
                −
                x
                ‖
              
              
            
            
              
              
                
                =
                
                  inf
                  
                    y
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                ‖
                ι
                −
                y
                ‖
                −
                
                  lim
                  
                    l
                    →
                    ∞
                  
                
                ‖
                ξ
                −
                
                  x
                  
                    l
                  
                
                ‖
              
              
            
            
              
              
                
                =
                
                  lim
                  
                    l
                    →
                    ∞
                  
                
                
                  inf
                  
                    y
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                
                  (
                  
                    ‖
                    ι
                    −
                    y
                    ‖
                    −
                    ‖
                    ξ
                    −
                    
                      x
                      
                        l
                      
                    
                    ‖
                  
                  )
                
              
              
            
            
              
              
                
                ≤
                
                  lim
                  
                    l
                    →
                    ∞
                  
                
                
                  inf
                  
                    y
                    ∈
                    
                      
                        R
                      
                      
                        d
                      
                    
                    ∖
                    O
                  
                
                
                  (
                  
                    ‖
                    ξ
                    −
                    ι
                    ‖
                    +
                    ‖
                    y
                    −
                    
                      x
                      
                        l
                      
                    
                    ‖
                  
                  )
                
              
              
                (
                ∗
                ∗
                )
                
                   with 
                
                x
                =
                
                  x
                  
                    l
                  
                
              
            
            
              
              
                
                =
                ‖
                ξ
                −
                ι
                ‖
              
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}|\mu (\xi )-\mu (\iota )|&=\mu (\iota )-\mu (\xi )&\\&=\inf _{y\in \mathbb {R} ^{d}\setminus O}\|\iota -y\|-\inf _{x\in \mathbb {R} ^{d}\setminus O}\|\xi -x\|&\\&=\inf _{y\in \mathbb {R} ^{d}\setminus O}\|\iota -y\|-\lim _{l\to \infty }\|\xi -x_{l}\|&\\&=\lim _{l\to \infty }\inf _{y\in \mathbb {R} ^{d}\setminus O}\left(\|\iota -y\|-\|\xi -x_{l}\|\right)&\\&\leq \lim _{l\to \infty }\inf _{y\in \mathbb {R} ^{d}\setminus O}\left(\|\xi -\iota \|+\|y-x_{l}\|\right)&(**){\text{ with }}x=x_{l}\\&=\|\xi -\iota \|&\end{aligned}}}
  Since always either 
  
    
      
        μ
        (
        ξ
        )
        ≥
        μ
        (
        ι
        )
      
    
    {\displaystyle \mu (\xi )\geq \mu (\iota )}
   or 
  
    
      
        μ
        (
        ξ
        )
        ≤
        μ
        (
        ι
        )
      
    
    {\displaystyle \mu (\xi )\leq \mu (\iota )}
  , we have proven Lipschitz continuity and thus continuity. By the extreme value theorem, 
  
    
      
        μ
      
    
    {\displaystyle \mu }
   therefore has a minimum 
  
    
      
        κ
        ∈
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle \kappa \in \mathbb {R} ^{d}}
  . Since 
  
    
      
        μ
        (
        κ
        )
        =
        0
      
    
    {\displaystyle \mu (\kappa )=0}
   would mean that 
  
    
      
        ‖
        ξ
        −
        
          x
          
            l
          
        
        ‖
        →
        0
        ,
        l
        →
        ∞
      
    
    {\displaystyle \|\xi -x_{l}\|\to 0,l\to \infty }
   for a sequence 
  
    
      
        (
        
          x
          
            l
          
        
        
          )
          
            l
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (x_{l})_{l\in \mathbb {N} }}
   in 
  
    
      
        
          
            R
          
          
            d
          
        
        ∖
        O
      
    
    {\displaystyle \mathbb {R} ^{d}\setminus O}
   which is a contradiction as 
  
    
      
        
          
            R
          
          
            d
          
        
        ∖
        O
      
    
    {\displaystyle \mathbb {R} ^{d}\setminus O}
   is closed and 
  
    
      
        κ
        ∈
        K
        ⊂
        O
      
    
    {\displaystyle \kappa \in K\subset O}
  , we have 
  
    
      
        μ
        (
        κ
        )
        >
        0
      
    
    {\displaystyle \mu (\kappa )>0}
  .
Hence, if we define 
  
    
      
        δ
        :=
        μ
        (
        κ
        )
      
    
    {\displaystyle \delta :=\mu (\kappa )}
  , then 
  
    
      
        δ
        >
        0
      
    
    {\displaystyle \delta >0}
  . Further, the function

  
    
      
        ϑ
        :
        
          
            R
          
          
            d
          
        
        →
        
          R
        
        ,
        ϑ
        (
        x
        )
        :=
        (
        
          χ
          
            K
            +
            
              B
              
                δ
                
                  /
                
                4
              
            
            (
            0
            )
          
        
        ∗
        
          η
          
            δ
            
              /
            
            4
          
        
        )
        (
        x
        )
        =
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        
          η
          
            δ
            
              /
            
            4
          
        
        (
        y
        )
        
          χ
          
            K
            +
            
              B
              
                δ
                
                  /
                
                4
              
            
            (
            0
            )
          
        
        (
        x
        −
        y
        )
        d
        y
        =
        
          ∫
          
            
              B
              
                δ
                
                  /
                
                4
              
            
            (
            0
            )
          
        
        
          η
          
            δ
            
              /
            
            4
          
        
        (
        y
        )
        
          χ
          
            K
            +
            
              B
              
                δ
                
                  /
                
                4
              
            
            (
            0
            )
          
        
        (
        x
        −
        y
        )
        d
        y
      
    
    {\displaystyle \vartheta :\mathbb {R} ^{d}\to \mathbb {R} ,\vartheta (x):=(\chi _{K+B_{\delta /4}(0)}*\eta _{\delta /4})(x)=\int _{\mathbb {R} ^{d}}\eta _{\delta /4}(y)\chi _{K+B_{\delta /4}(0)}(x-y)dy=\int _{B_{\delta /4}(0)}\eta _{\delta /4}(y)\chi _{K+B_{\delta /4}(0)}(x-y)dy}
  has support contained in 
  
    
      
        O
      
    
    {\displaystyle O}
  , is equal to 
  
    
      
        1
      
    
    {\displaystyle 1}
   within 
  
    
      
        K
      
    
    {\displaystyle K}
   and further is contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(\mathbb {R} ^{d})}
   due to lemma 4.7. Hence, it is also contained in 
  
    
      
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}(O)}
  . Since therefore, by the monotonicity of the integral

  
    
      
        
          ∫
          
            K
          
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        d
        x
        =
        
          ∫
          
            O
          
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        
          χ
          
            K
          
        
        (
        x
        )
        d
        x
        ≤
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        ϑ
        (
        x
        )
        d
        x
      
    
    {\displaystyle \int _{K}|f(x)|dx=\int _{O}|f(x)|\chi _{K}(x)dx\leq \int _{\mathbb {R} ^{d}}|f(x)|\vartheta (x)dx}
  , 
  
    
      
        f
      
    
    {\displaystyle f}
   is indeed locally integrable.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  

Proof:
From Hölder's inequality we obtain

  
    
      
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        
          |
        
        ϕ
        (
        x
        )
        
          |
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        d
        x
        ≤
        ‖
        ϕ
        
          ‖
          
            
              L
              
                2
              
            
          
        
        ‖
        f
        
          ‖
          
            
              L
              
                2
              
            
          
        
        <
        ∞
      
    
    {\displaystyle \int _{\mathbb {R} ^{d}}|\phi (x)||f(x)|dx\leq \|\phi \|_{L^{2}}\|f\|_{L^{2}}<\infty }
  .Hence, 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
      
    
    {\displaystyle {\mathcal {T}}_{f}}
   is well-defined.
Due to the triangle inequality for integrals and Hölder's inequality, we have

  
    
      
        
          |
        
        
          T
          
            f
          
        
        (
        
          ϕ
          
            l
          
        
        )
        −
        
          T
          
            f
          
        
        (
        ϕ
        )
        
          |
        
        ≤
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        
          |
        
        (
        
          ϕ
          
            l
          
        
        −
        ϕ
        )
        (
        x
        )
        
          |
        
        
          |
        
        f
        (
        x
        )
        
          |
        
        d
        x
        ≤
        ‖
        
          ϕ
          
            l
          
        
        −
        ϕ
        
          ‖
          
            
              L
              
                2
              
            
          
        
        ‖
        f
        
          ‖
          
            
              L
              
                2
              
            
          
        
      
    
    {\displaystyle |T_{f}(\phi _{l})-T_{f}(\phi )|\leq \int _{\mathbb {R} ^{d}}|(\phi _{l}-\phi )(x)||f(x)|dx\leq \|\phi _{l}-\phi \|_{L^{2}}\|f\|_{L^{2}}}
  Furthermore

  
    
      
        
          
            
              
                ‖
                
                  ϕ
                  
                    l
                  
                
                −
                ϕ
                
                  ‖
                  
                    
                      L
                      
                        2
                      
                    
                  
                  
                    2
                  
                
              
              
                
                ≤
                ‖
                
                  ϕ
                  
                    l
                  
                
                −
                ϕ
                
                  ‖
                  
                    ∞
                  
                
                
                  ∫
                  
                    
                      
                        R
                      
                      
                        d
                      
                    
                  
                
                
                  |
                
                (
                
                  ϕ
                  
                    l
                  
                
                −
                ϕ
                )
                (
                x
                )
                
                  |
                
                d
                x
              
            
            
              
              
                
                =
                ‖
                
                  ϕ
                  
                    l
                  
                
                −
                ϕ
                
                  ‖
                  
                    ∞
                  
                
                
                  ∫
                  
                    
                      
                        R
                      
                      
                        d
                      
                    
                  
                
                
                  ∏
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                (
                1
                +
                
                  x
                  
                    j
                  
                  
                    2
                  
                
                )
                
                  |
                
                (
                
                  ϕ
                  
                    l
                  
                
                −
                ϕ
                )
                (
                x
                )
                
                  |
                
                
                  
                    1
                    
                      
                        ∏
                        
                          j
                          =
                          1
                        
                        
                          d
                        
                      
                      (
                      1
                      +
                      
                        x
                        
                          j
                        
                        
                          2
                        
                      
                      )
                    
                  
                
                d
                x
              
            
            
              
              
                
                ≤
                ‖
                
                  ϕ
                  
                    l
                  
                
                −
                ϕ
                
                  ‖
                  
                    ∞
                  
                
                
                  
                    ‖
                    
                      
                        ∏
                        
                          j
                          =
                          1
                        
                        
                          d
                        
                      
                      (
                      1
                      +
                      
                        x
                        
                          j
                        
                        
                          2
                        
                      
                      )
                      (
                      
                        ϕ
                        
                          l
                        
                      
                      −
                      ϕ
                      )
                    
                    ‖
                  
                  
                    ∞
                  
                
                
                  
                    
                      
                        
                          ∫
                          
                            
                              
                                R
                              
                              
                                d
                              
                            
                          
                        
                        
                          
                            1
                            
                              
                                ∏
                                
                                  j
                                  =
                                  1
                                
                                
                                  d
                                
                              
                              (
                              1
                              +
                              
                                x
                                
                                  j
                                
                                
                                  2
                                
                              
                              )
                            
                          
                        
                        d
                        x
                      
                      ⏟
                    
                  
                  
                    =
                    
                      π
                      
                        d
                      
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\|\phi _{l}-\phi \|_{L^{2}}^{2}&\leq \|\phi _{l}-\phi \|_{\infty }\int _{\mathbb {R} ^{d}}|(\phi _{l}-\phi )(x)|dx\\&=\|\phi _{l}-\phi \|_{\infty }\int _{\mathbb {R} ^{d}}\prod _{j=1}^{d}(1+x_{j}^{2})|(\phi _{l}-\phi )(x)|{\frac {1}{\prod _{j=1}^{d}(1+x_{j}^{2})}}dx\\&\leq \|\phi _{l}-\phi \|_{\infty }\left\|\prod _{j=1}^{d}(1+x_{j}^{2})(\phi _{l}-\phi )\right\|_{\infty }\underbrace {\int _{\mathbb {R} ^{d}}{\frac {1}{\prod _{j=1}^{d}(1+x_{j}^{2})}}dx} _{=\pi ^{d}}\end{aligned}}}
  .If 
  
    
      
        
          ϕ
          
            l
          
        
        →
        ϕ
      
    
    {\displaystyle \phi _{l}\to \phi }
   in the notion of convergence of the Schwartz function space, then this expression goes to zero. Therefore, continuity is verified.
Linearity follows from the linearity of the integral.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Equicontinuity ==
We now introduce the concept of equicontinuity.

So equicontinuity is in fact defined for sets of continuous functions mapping from 
  
    
      
        X
      
    
    {\displaystyle X}
   (a set in a metric space) to the real numbers 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  .

Proof:
In order to prove uniform convergence, by definition we must prove that for all 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
  , there exists an 
  
    
      
        N
        ∈
        
          N
        
      
    
    {\displaystyle N\in \mathbb {N} }
   such that for all 
  
    
      
        l
        ≥
        N
        :
        ∀
        x
        ∈
        Q
        :
        
          |
        
        
          f
          
            l
          
        
        (
        x
        )
        −
        f
        (
        x
        )
        
          |
        
        <
        ϵ
      
    
    {\displaystyle l\geq N:\forall x\in Q:|f_{l}(x)-f(x)|<\epsilon }
  .
So let's assume the contrary, which equals by negating the logical statement

  
    
      
        ∃
        ϵ
        >
        0
        :
        ∀
        N
        ∈
        
          N
        
        :
        ∃
        l
        ≥
        N
        :
        ∃
        x
        ∈
        Q
        :
        
          |
        
        
          f
          
            l
          
        
        (
        x
        )
        −
        f
        (
        x
        )
        
          |
        
        ≥
        ϵ
      
    
    {\displaystyle \exists \epsilon >0:\forall N\in \mathbb {N} :\exists l\geq N:\exists x\in Q:|f_{l}(x)-f(x)|\geq \epsilon }
  .We choose a sequence 
  
    
      
        (
        
          x
          
            m
          
        
        
          )
          
            m
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (x_{m})_{m\in \mathbb {N} }}
   in 
  
    
      
        Q
      
    
    {\displaystyle Q}
  . We take 
  
    
      
        
          x
          
            1
          
        
      
    
    {\displaystyle x_{1}}
   in 
  
    
      
        Q
      
    
    {\displaystyle Q}
   such that 
  
    
      
        
          |
        
        
          f
          
            
              l
              
                1
              
            
          
        
        (
        
          x
          
            1
          
        
        )
        −
        f
        (
        
          x
          
            1
          
        
        )
        
          |
        
        ≥
        ϵ
      
    
    {\displaystyle |f_{l_{1}}(x_{1})-f(x_{1})|\geq \epsilon }
   for an arbitrarily chosen 
  
    
      
        
          l
          
            1
          
        
        ∈
        
          N
        
      
    
    {\displaystyle l_{1}\in \mathbb {N} }
   and if we have already chosen 
  
    
      
        
          x
          
            k
          
        
      
    
    {\displaystyle x_{k}}
   and 
  
    
      
        
          l
          
            k
          
        
      
    
    {\displaystyle l_{k}}
   for all 
  
    
      
        k
        ∈
        {
        1
        ,
        …
        ,
        m
        }
      
    
    {\displaystyle k\in \{1,\ldots ,m\}}
  , we choose 
  
    
      
        
          x
          
            m
            +
            1
          
        
      
    
    {\displaystyle x_{m+1}}
   such that 
  
    
      
        
          |
        
        
          f
          
            
              l
              
                m
                +
                1
              
            
          
        
        (
        
          x
          
            m
            +
            1
          
        
        )
        −
        f
        (
        
          x
          
            m
            +
            1
          
        
        )
        
          |
        
        ≥
        ϵ
      
    
    {\displaystyle |f_{l_{m+1}}(x_{m+1})-f(x_{m+1})|\geq \epsilon }
  , where 
  
    
      
        
          l
          
            m
            +
            1
          
        
      
    
    {\displaystyle l_{m+1}}
   is greater than 
  
    
      
        
          l
          
            m
          
        
      
    
    {\displaystyle l_{m}}
  .
As 
  
    
      
        Q
      
    
    {\displaystyle Q}
   is sequentially compact, there is a convergent subsequence 
  
    
      
        (
        
          x
          
            
              m
              
                j
              
            
          
        
        
          )
          
            j
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (x_{m_{j}})_{j\in \mathbb {N} }}
   of 
  
    
      
        (
        
          x
          
            m
          
        
        
          )
          
            m
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (x_{m})_{m\in \mathbb {N} }}
  . Let us call the limit of that subsequence sequence 
  
    
      
        x
      
    
    {\displaystyle x}
  .
As 
  
    
      
        
          
            Q
          
        
      
    
    {\displaystyle {\mathcal {Q}}}
   is equicontinuous, we can choose 
  
    
      
        δ
        ∈
        
          
            R
          
          
            >
            0
          
        
      
    
    {\displaystyle \delta \in \mathbb {R} _{>0}}
   such that 

  
    
      
        ‖
        x
        −
        y
        ‖
        <
        δ
        ⇒
        ∀
        f
        ∈
        
          
            Q
          
        
        :
        
          |
        
        f
        (
        x
        )
        −
        f
        (
        y
        )
        
          |
        
        <
        
          
            ϵ
            4
          
        
      
    
    {\displaystyle \|x-y\|<\delta \Rightarrow \forall f\in {\mathcal {Q}}:|f(x)-f(y)|<{\frac {\epsilon }{4}}}
  .Further, since 
  
    
      
        
          x
          
            
              m
              
                j
              
            
          
        
        →
        x
      
    
    {\displaystyle x_{m_{j}}\to x}
   (if 
  
    
      
        j
        →
        ∞
      
    
    {\displaystyle j\to \infty }
   of course), we may choose 
  
    
      
        J
        ∈
        
          N
        
      
    
    {\displaystyle J\in \mathbb {N} }
   such that

  
    
      
        ∀
        j
        ≥
        J
        :
        ‖
        
          x
          
            
              m
              
                j
              
            
          
        
        −
        x
        ‖
        <
        δ
      
    
    {\displaystyle \forall j\geq J:\|x_{m_{j}}-x\|<\delta }
  .But then follows for 
  
    
      
        j
        ≥
        J
      
    
    {\displaystyle j\geq J}
   and the reverse triangle inequality:

  
    
      
        
          |
        
        
          f
          
            
              l
              
                
                  m
                  
                    j
                  
                
              
            
          
        
        (
        x
        )
        −
        f
        (
        x
        )
        
          |
        
        ≥
        
          |
          
            
              |
            
            
              f
              
                
                  l
                  
                    
                      m
                      
                        j
                      
                    
                  
                
              
            
            (
            x
            )
            −
            f
            (
            
              x
              
                
                  m
                  
                    j
                  
                
              
            
            )
            
              |
            
            −
            
              |
            
            f
            (
            
              x
              
                
                  m
                  
                    j
                  
                
              
            
            )
            −
            f
            (
            x
            )
            
              |
            
          
          |
        
      
    
    {\displaystyle |f_{l_{m_{j}}}(x)-f(x)|\geq \left||f_{l_{m_{j}}}(x)-f(x_{m_{j}})|-|f(x_{m_{j}})-f(x)|\right|}
  Since we had 
  
    
      
        
          |
        
        f
        (
        
          x
          
            
              m
              
                j
              
            
          
        
        )
        −
        f
        (
        x
        )
        
          |
        
        <
        
          
            ϵ
            4
          
        
      
    
    {\displaystyle |f(x_{m_{j}})-f(x)|<{\frac {\epsilon }{4}}}
  , the reverse triangle inequality and the definition of t

  
    
      
        
          |
        
        
          f
          
            
              l
              
                
                  m
                  
                    j
                  
                
              
            
          
        
        (
        x
        )
        −
        f
        (
        
          x
          
            
              m
              
                j
              
            
          
        
        )
        
          |
        
        ≥
        
          |
          
            
              |
            
            
              f
              
                
                  l
                  
                    
                      m
                      
                        j
                      
                    
                  
                
              
            
            (
            
              x
              
                
                  m
                  
                    j
                  
                
              
            
            )
            −
            f
            (
            
              x
              
                
                  m
                  
                    j
                  
                
              
            
            )
            
              |
            
            −
            
              |
            
            
              f
              
                
                  l
                  
                    
                      m
                      
                        j
                      
                    
                  
                
              
            
            (
            x
            )
            −
            
              f
              
                
                  l
                  
                    
                      m
                      
                        j
                      
                    
                  
                
              
            
            (
            
              x
              
                
                  m
                  
                    j
                  
                
              
            
            )
            
              |
            
          
          |
        
        ≥
        ϵ
        −
        
          
            ϵ
            4
          
        
      
    
    {\displaystyle |f_{l_{m_{j}}}(x)-f(x_{m_{j}})|\geq \left||f_{l_{m_{j}}}(x_{m_{j}})-f(x_{m_{j}})|-|f_{l_{m_{j}}}(x)-f_{l_{m_{j}}}(x_{m_{j}})|\right|\geq \epsilon -{\frac {\epsilon }{4}}}
  , we obtain:

  
    
      
        
          
            
              
                
                  |
                
                
                  f
                  
                    
                      l
                      
                        
                          m
                          
                            j
                          
                        
                      
                    
                  
                
                (
                x
                )
                −
                f
                (
                x
                )
                
                  |
                
              
              
                
                ≥
                
                  |
                  
                    
                      |
                    
                    
                      f
                      
                        
                          l
                          
                            
                              m
                              
                                j
                              
                            
                          
                        
                      
                    
                    (
                    x
                    )
                    −
                    f
                    (
                    
                      x
                      
                        
                          m
                          
                            j
                          
                        
                      
                    
                    )
                    
                      |
                    
                    −
                    
                      |
                    
                    f
                    (
                    
                      x
                      
                        
                          m
                          
                            j
                          
                        
                      
                    
                    )
                    −
                    f
                    (
                    x
                    )
                    
                      |
                    
                  
                  |
                
              
            
            
              
              
                
                =
                
                  |
                
                
                  f
                  
                    
                      l
                      
                        
                          m
                          
                            j
                          
                        
                      
                    
                  
                
                (
                x
                )
                −
                f
                (
                
                  x
                  
                    
                      m
                      
                        j
                      
                    
                  
                
                )
                
                  |
                
                −
                
                  |
                
                f
                (
                
                  x
                  
                    
                      m
                      
                        j
                      
                    
                  
                
                )
                −
                f
                (
                x
                )
                
                  |
                
              
            
            
              
              
                
                ≥
                ϵ
                −
                
                  
                    ϵ
                    4
                  
                
                −
                
                  
                    ϵ
                    4
                  
                
              
            
            
              
              
                
                ≥
                
                  
                    ϵ
                    2
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}|f_{l_{m_{j}}}(x)-f(x)|&\geq \left||f_{l_{m_{j}}}(x)-f(x_{m_{j}})|-|f(x_{m_{j}})-f(x)|\right|\\&=|f_{l_{m_{j}}}(x)-f(x_{m_{j}})|-|f(x_{m_{j}})-f(x)|\\&\geq \epsilon -{\frac {\epsilon }{4}}-{\frac {\epsilon }{4}}\\&\geq {\frac {\epsilon }{2}}\end{aligned}}}
  Thus we have a contradiction to 
  
    
      
        
          f
          
            l
          
        
        (
        x
        )
        →
        f
        (
        x
        )
      
    
    {\displaystyle f_{l}(x)\to f(x)}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  

Proof: We have to prove equicontinuity, so we have to prove

  
    
      
        ∀
        x
        ∈
        X
        :
        ∃
        δ
        ∈
        
          
            R
          
          
            >
            0
          
        
        :
        ∀
        y
        ∈
        X
        :
        ‖
        x
        −
        y
        ‖
        <
        δ
        ⇒
        ∀
        f
        ∈
        
          
            Q
          
        
        :
        
          |
        
        f
        (
        x
        )
        −
        f
        (
        y
        )
        
          |
        
        <
        ϵ
      
    
    {\displaystyle \forall x\in X:\exists \delta \in \mathbb {R} _{>0}:\forall y\in X:\|x-y\|<\delta \Rightarrow \forall f\in {\mathcal {Q}}:|f(x)-f(y)|<\epsilon }
  .Let 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
   be arbitrary.
We choose 
  
    
      
        δ
        :=
        
          
            ϵ
            b
          
        
      
    
    {\displaystyle \delta :={\frac {\epsilon }{b}}}
  .
Let 
  
    
      
        y
        ∈
        X
      
    
    {\displaystyle y\in X}
   such that 
  
    
      
        ‖
        x
        −
        y
        ‖
        <
        δ
      
    
    {\displaystyle \|x-y\|<\delta }
  , and let 
  
    
      
        f
        ∈
        
          
            Q
          
        
      
    
    {\displaystyle f\in {\mathcal {Q}}}
   be arbitrary. By the mean-value theorem in multiple dimensions, we obtain that there exists a 
  
    
      
        λ
        ∈
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle \lambda \in [0,1]}
   such that:

  
    
      
        f
        (
        x
        )
        −
        f
        (
        y
        )
        =
        ∇
        f
        (
        λ
        x
        +
        (
        1
        −
        λ
        )
        y
        )
        ⋅
        (
        x
        −
        y
        )
      
    
    {\displaystyle f(x)-f(y)=\nabla f(\lambda x+(1-\lambda )y)\cdot (x-y)}
  The element 
  
    
      
        λ
        x
        +
        (
        1
        −
        λ
        )
        y
      
    
    {\displaystyle \lambda x+(1-\lambda )y}
   is inside 
  
    
      
        X
      
    
    {\displaystyle X}
  , because 
  
    
      
        X
      
    
    {\displaystyle X}
   is convex. From the Cauchy-Schwarz inequality then follows:

  
    
      
        
          |
        
        f
        (
        x
        )
        −
        f
        (
        y
        )
        
          |
        
        =
        
          |
        
        ∇
        f
        (
        λ
        x
        +
        (
        1
        −
        λ
        )
        y
        )
        ⋅
        (
        x
        −
        y
        )
        
          |
        
        ≤
        ‖
        ∇
        f
        (
        λ
        x
        +
        (
        1
        −
        λ
        )
        y
        )
        ‖
        ‖
        x
        −
        y
        ‖
        <
        b
        δ
        =
        
          
            b
            b
          
        
        ϵ
        =
        ϵ
      
    
    {\displaystyle |f(x)-f(y)|=|\nabla f(\lambda x+(1-\lambda )y)\cdot (x-y)|\leq \|\nabla f(\lambda x+(1-\lambda )y)\|\|x-y\|
        0
      
    
    {\displaystyle \alpha _{k}>0}
   (we may do this because 
  
    
      
        
          |
        
        α
        
          |
        
        =
        k
        +
        1
        >
        0
      
    
    {\displaystyle |\alpha |=k+1>0}
  ). We define again 
  
    
      
        
          e
          
            k
          
        
        =
        (
        0
        ,
        …
        ,
        0
        ,
        1
        ,
        0
        ,
        …
        ,
        0
        )
      
    
    {\displaystyle e_{k}=(0,\ldots ,0,1,0,\ldots ,0)}
  , where the 
  
    
      
        1
      
    
    {\displaystyle 1}
   is at the 
  
    
      
        k
      
    
    {\displaystyle k}
  -th place. Due to Schwarz' theorem and the ordinary product rule, we have

  
    
      
        
          ∂
          
            α
          
        
        f
        g
        =
        
          ∂
          
            α
            −
            
              e
              
                k
              
            
          
        
        
          (
          
            
              ∂
              
                
                  x
                  
                    k
                  
                
              
            
            f
            g
          
          )
        
        =
        
          ∂
          
            α
            −
            
              e
              
                k
              
            
          
        
        
          (
          
            
              ∂
              
                
                  x
                  
                    k
                  
                
              
            
            f
            g
            +
            f
            
              ∂
              
                
                  x
                  
                    k
                  
                
              
            
            g
          
          )
        
      
    
    {\displaystyle \partial _{\alpha }fg=\partial _{\alpha -e_{k}}\left(\partial _{x_{k}}fg\right)=\partial _{\alpha -e_{k}}\left(\partial _{x_{k}}fg+f\partial _{x_{k}}g\right)}
  .By linearity of derivatives and induction hypothesis, we have

  
    
      
        
          
            
              
                
                  ∂
                  
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                  
                
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            k
                          
                        
                      
                    
                    f
                    g
                    +
                    f
                    
                      ∂
                      
                        
                          x
                          
                            k
                          
                        
                      
                    
                    g
                  
                  )
                
              
              
                
                =
                
                  ∂
                  
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                  
                
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            k
                          
                        
                      
                    
                    f
                    g
                  
                  )
                
                +
                
                  ∂
                  
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                  
                
                
                  (
                  
                    f
                    
                      ∂
                      
                        
                          x
                          
                            k
                          
                        
                      
                    
                    g
                  
                  )
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    ς
                    ≤
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                  
                
                
                  
                    
                      (
                    
                    
                      
                        α
                        −
                        
                          e
                          
                            k
                          
                        
                      
                      ς
                    
                    
                      )
                    
                  
                
                
                  ∂
                  
                    ς
                  
                
                
                  ∂
                  
                    
                      x
                      
                        k
                      
                    
                  
                
                f
                
                  ∂
                  
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                    −
                    ς
                  
                
                g
                +
                
                  ∑
                  
                    ς
                    ≤
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                  
                
                
                  
                    
                      (
                    
                    
                      
                        α
                        −
                        
                          e
                          
                            k
                          
                        
                      
                      ς
                    
                    
                      )
                    
                  
                
                
                  ∂
                  
                    ς
                  
                
                f
                
                  ∂
                  
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                    −
                    ς
                  
                
                
                  ∂
                  
                    
                      x
                      
                        k
                      
                    
                  
                
                g
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\partial _{\alpha -e_{k}}\left(\partial _{x_{k}}fg+f\partial _{x_{k}}g\right)&=\partial _{\alpha -e_{k}}\left(\partial _{x_{k}}fg\right)+\partial _{\alpha -e_{k}}\left(f\partial _{x_{k}}g\right)\\&=\sum _{\varsigma \leq \alpha -e_{k}}{\binom {\alpha -e_{k}}{\varsigma }}\partial _{\varsigma }\partial _{x_{k}}f\partial _{\alpha -e_{k}-\varsigma }g+\sum _{\varsigma \leq \alpha -e_{k}}{\binom {\alpha -e_{k}}{\varsigma }}\partial _{\varsigma }f\partial _{\alpha -e_{k}-\varsigma }\partial _{x_{k}}g\end{aligned}}}
  .Since

  
    
      
        
          ∂
          
            α
            −
            
              e
              
                k
              
            
            −
            ς
          
        
        =
        
          ∂
          
            α
            −
            (
            ς
            +
            
              e
              
                k
              
            
            )
          
        
      
    
    {\displaystyle \partial _{\alpha -e_{k}-\varsigma }=\partial _{\alpha -(\varsigma +e_{k})}}
  and

  
    
      
        {
        ς
        ∈
        
          
            N
          
          
            0
          
          
            d
          
        
        
          |
        
        0
        ≤
        ς
        ≤
        α
        −
        
          e
          
            k
          
        
        }
        =
        {
        ς
        −
        
          e
          
            k
          
        
        ∈
        
          
            N
          
          
            0
          
          
            d
          
        
        
          |
        
        
          e
          
            k
          
        
        ≤
        ς
        ≤
        α
        }
      
    
    {\displaystyle \{\varsigma \in \mathbb {N} _{0}^{d}|0\leq \varsigma \leq \alpha -e_{k}\}=\{\varsigma -e_{k}\in \mathbb {N} _{0}^{d}|e_{k}\leq \varsigma \leq \alpha \}}
  ,we are allowed to shift indices in the first of the two above sums, and furthermore we have by definition

  
    
      
        
          ∂
          
            ς
          
        
        
          ∂
          
            
              x
              
                k
              
            
          
        
        =
        
          ∂
          
            ς
            +
            
              e
              
                k
              
            
          
        
      
    
    {\displaystyle \partial _{\varsigma }\partial _{x_{k}}=\partial _{\varsigma +e_{k}}}
  .With this, we obtain

  
    
      
        
          ∑
          
            ς
            ≤
            α
            −
            
              e
              
                k
              
            
          
        
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    k
                  
                
              
              ς
            
            
              )
            
          
        
        
          ∂
          
            ς
          
        
        
          ∂
          
            
              x
              
                k
              
            
          
        
        f
        
          ∂
          
            α
            −
            
              e
              
                k
              
            
            −
            ς
          
        
        g
        +
        
          ∑
          
            ς
            ≤
            α
            −
            
              e
              
                k
              
            
          
        
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    k
                  
                
              
              ς
            
            
              )
            
          
        
        
          ∂
          
            ς
          
        
        f
        
          ∂
          
            α
            −
            
              e
              
                k
              
            
            −
            ς
          
        
        
          ∂
          
            
              x
              
                k
              
            
          
        
        g
        =
        
          ∑
          
            
              e
              
                k
              
            
            ≤
            ς
            ≤
            α
          
        
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    k
                  
                
              
              
                ς
                −
                
                  e
                  
                    k
                  
                
              
            
            
              )
            
          
        
        
          ∂
          
            ς
          
        
        f
        ⋅
        
          ∂
          
            α
            −
            ς
          
        
        g
        +
        
          ∑
          
            ς
            ≤
            α
            −
            
              e
              
                k
              
            
          
        
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    k
                  
                
              
              ς
            
            
              )
            
          
        
        
          ∂
          
            ς
          
        
        f
        
          ∂
          
            α
            −
            ς
          
        
        g
      
    
    {\displaystyle \sum _{\varsigma \leq \alpha -e_{k}}{\binom {\alpha -e_{k}}{\varsigma }}\partial _{\varsigma }\partial _{x_{k}}f\partial _{\alpha -e_{k}-\varsigma }g+\sum _{\varsigma \leq \alpha -e_{k}}{\binom {\alpha -e_{k}}{\varsigma }}\partial _{\varsigma }f\partial _{\alpha -e_{k}-\varsigma }\partial _{x_{k}}g=\sum _{e_{k}\leq \varsigma \leq \alpha }{\binom {\alpha -e_{k}}{\varsigma -e_{k}}}\partial _{\varsigma }f\cdot \partial _{\alpha -\varsigma }g+\sum _{\varsigma \leq \alpha -e_{k}}{\binom {\alpha -e_{k}}{\varsigma }}\partial _{\varsigma }f\partial _{\alpha -\varsigma }g}
  Due to lemma 4.18,

  
    
      
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    k
                  
                
              
              
                β
                −
                
                  e
                  
                    i
                  
                
              
            
            
              )
            
          
        
        +
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    k
                  
                
              
              β
            
            
              )
            
          
        
        =
        
          
            
              (
            
            
              α
              β
            
            
              )
            
          
        
      
    
    {\displaystyle {\binom {\alpha -e_{k}}{\beta -e_{i}}}+{\binom {\alpha -e_{k}}{\beta }}={\binom {\alpha }{\beta }}}
  .Further, we have

  
    
      
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    i
                  
                
              
              0
            
            
              )
            
          
        
        =
        
          
            
              (
            
            
              α
              0
            
            
              )
            
          
        
        =
        1
      
    
    {\displaystyle {\binom {\alpha -e_{i}}{0}}={\binom {\alpha }{0}}=1}
   where 
  
    
      
        0
        =
        (
        0
        ,
        …
        ,
        0
        )
      
    
    {\displaystyle 0=(0,\ldots ,0)}
   in 
  
    
      
        
          
            N
          
          
            0
          
          
            d
          
        
      
    
    {\displaystyle \mathbb {N} _{0}^{d}}
  ,and

  
    
      
        
          
            
              (
            
            
              
                α
                −
                
                  e
                  
                    k
                  
                
              
              
                α
                −
                
                  e
                  
                    k
                  
                
              
            
            
              )
            
          
        
        =
        
          
            
              (
            
            
              α
              α
            
            
              )
            
          
        
        =
        1
      
    
    {\displaystyle {\binom {\alpha -e_{k}}{\alpha -e_{k}}}={\binom {\alpha }{\alpha }}=1}
  (these two rules may be checked from the definition of 
  
    
      
        
          
            
              (
            
            
              α
              β
            
            
              )
            
          
        
      
    
    {\displaystyle {\binom {\alpha }{\beta }}}
  ). It follows

  
    
      
        
          
            
              
                
                  ∂
                  
                    α
                  
                
                (
                f
                g
                )
              
              
                
                =
                
                  ∑
                  
                    
                      e
                      
                        k
                      
                    
                    ≤
                    ς
                    ≤
                    α
                  
                
                
                  
                    
                      (
                    
                    
                      
                        α
                        −
                        
                          e
                          
                            k
                          
                        
                      
                      
                        ς
                        −
                        
                          e
                          
                            k
                          
                        
                      
                    
                    
                      )
                    
                  
                
                
                  ∂
                  
                    ς
                  
                
                f
                ⋅
                
                  ∂
                  
                    α
                    −
                    ς
                  
                
                g
                +
                
                  ∑
                  
                    ς
                    ≤
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                  
                
                
                  
                    
                      (
                    
                    
                      
                        α
                        −
                        
                          e
                          
                            k
                          
                        
                      
                      ς
                    
                    
                      )
                    
                  
                
                
                  ∂
                  
                    ς
                  
                
                f
                
                  ∂
                  
                    α
                    −
                    ς
                  
                
                g
              
            
            
              
              
                
                =
                
                  
                    
                      (
                    
                    
                      
                        α
                        −
                        
                          e
                          
                            k
                          
                        
                      
                      0
                    
                    
                      )
                    
                  
                
                f
                
                  ∂
                  
                    α
                  
                
                g
                +
                
                  ∑
                  
                    
                      e
                      
                        k
                      
                    
                    ≤
                    ς
                    ≤
                    α
                    −
                    
                      e
                      
                        k
                      
                    
                  
                
                
                  [
                  
                    
                      
                        
                          (
                        
                        
                          
                            α
                            −
                            
                              e
                              
                                k
                              
                            
                          
                          
                            ς
                            −
                            
                              e
                              
                                k
                              
                            
                          
                        
                        
                          )
                        
                      
                    
                    +
                    
                      
                        
                          (
                        
                        
                          
                            α
                            −
                            
                              e
                              
                                k
                              
                            
                          
                          ς
                        
                        
                          )
                        
                      
                    
                  
                  ]
                
                
                  ∂
                  
                    ς
                  
                
                f
                
                  ∂
                  
                    α
                    −
                    ς
                  
                
                g
                +
                
                  
                    
                      (
                    
                    
                      
                        α
                        −
                        
                          e
                          
                            k
                          
                        
                      
                      
                        α
                        −
                        
                          e
                          
                            k
                          
                        
                      
                    
                    
                      )
                    
                  
                
                f
                
                  ∂
                  
                    α
                  
                
                g
              
            
            
              
              
                
                =
                
                  ∑
                  
                    ς
                    ≤
                    α
                  
                
                
                  
                    
                      (
                    
                    
                      α
                      ς
                    
                    
                      )
                    
                  
                
                
                  ∂
                  
                    ς
                  
                
                f
                
                  ∂
                  
                    α
                    −
                    ς
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\partial _{\alpha }(fg)&=\sum _{e_{k}\leq \varsigma \leq \alpha }{\binom {\alpha -e_{k}}{\varsigma -e_{k}}}\partial _{\varsigma }f\cdot \partial _{\alpha -\varsigma }g+\sum _{\varsigma \leq \alpha -e_{k}}{\binom {\alpha -e_{k}}{\varsigma }}\partial _{\varsigma }f\partial _{\alpha -\varsigma }g\\&={\binom {\alpha -e_{k}}{0}}f\partial _{\alpha }g+\sum _{e_{k}\leq \varsigma \leq \alpha -e_{k}}\left[{\binom {\alpha -e_{k}}{\varsigma -e_{k}}}+{\binom {\alpha -e_{k}}{\varsigma }}\right]\partial _{\varsigma }f\partial _{\alpha -\varsigma }g+{\binom {\alpha -e_{k}}{\alpha -e_{k}}}f\partial _{\alpha }g\\&=\sum _{\varsigma \leq \alpha }{\binom {\alpha }{\varsigma }}\partial _{\varsigma }f\partial _{\alpha -\varsigma }\end{aligned}}}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Operations on Distributions ==
For 
  
    
      
        φ
        ,
        ϑ
        ∈
        
          
            D
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle \varphi ,\vartheta \in {\mathcal {D}}(\mathbb {R} ^{d})}
   there are operations such as the differentiation of 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  , the convolution of 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   and 
  
    
      
        ϑ
      
    
    {\displaystyle \vartheta }
   and the multiplication of 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   and 
  
    
      
        ϑ
      
    
    {\displaystyle \vartheta }
  . In the following section, we want to define these three operations (differentiation, convolution with 
  
    
      
        ϑ
      
    
    {\displaystyle \vartheta }
   and multiplication with 
  
    
      
        ϑ
      
    
    {\displaystyle \vartheta }
  ) for a distribution 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
   instead of 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  .

Proof:
We have to prove two claims: First, that the function 
  
    
      
        φ
        ↦
        
          
            T
          
        
        (
        
          
            L
          
        
        (
        φ
        )
        )
      
    
    {\displaystyle \varphi \mapsto {\mathcal {T}}({\mathcal {L}}(\varphi ))}
   is a distribution, and second that 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
   as defined above has the property

  
    
      
        ∀
        φ
        ∈
        
          
            D
          
        
        (
        O
        )
        :
        Λ
        (
        
          
            
              T
            
          
          
            φ
          
        
        )
        =
        
          
            
              T
            
          
          
            L
            (
            φ
            )
          
        
      
    
    {\displaystyle \forall \varphi \in {\mathcal {D}}(O):\Lambda ({\mathcal {T}}_{\varphi })={\mathcal {T}}_{L(\varphi )}}
  1.
We show that the function 
  
    
      
        φ
        ↦
        
          
            T
          
        
        (
        
          
            L
          
        
        (
        φ
        )
        )
      
    
    {\displaystyle \varphi \mapsto {\mathcal {T}}({\mathcal {L}}(\varphi ))}
   is a distribution.

  
    
      
        
          
            T
          
        
        (
        
          
            L
          
        
        (
        φ
        )
        )
      
    
    {\displaystyle {\mathcal {T}}({\mathcal {L}}(\varphi ))}
   has a well-defined value in 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
   as 
  
    
      
        
          
            L
          
        
      
    
    {\displaystyle {\mathcal {L}}}
   maps to 
  
    
      
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}(O)}
  , which is exactly the preimage of 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
  . The function 
  
    
      
        φ
        ↦
        
          
            T
          
        
        (
        
          
            L
          
        
        (
        φ
        )
        )
      
    
    {\displaystyle \varphi \mapsto {\mathcal {T}}({\mathcal {L}}(\varphi ))}
   is continuous since it is the composition of two continuous functions, and it is linear for the same reason (see exercise 2).
2.
We show that 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
   has the property

  
    
      
        ∀
        φ
        ∈
        
          
            D
          
        
        (
        O
        )
        :
        Λ
        (
        
          
            
              T
            
          
          
            φ
          
        
        )
        =
        
          
            
              T
            
          
          
            L
            (
            φ
            )
          
        
      
    
    {\displaystyle \forall \varphi \in {\mathcal {D}}(O):\Lambda ({\mathcal {T}}_{\varphi })={\mathcal {T}}_{L(\varphi )}}
  For every 
  
    
      
        ϑ
        ∈
        
          
            D
          
        
        (
        U
        )
      
    
    {\displaystyle \vartheta \in {\mathcal {D}}(U)}
  , we have

  
    
      
        Λ
        (
        
          
            
              T
            
          
          
            φ
          
        
        )
        (
        ϑ
        )
        :=
        (
        
          
            
              T
            
          
          
            φ
          
        
        ∘
        
          
            L
          
        
        )
        (
        ϑ
        )
        :=
        
          ∫
          
            O
          
        
        φ
        (
        x
        )
        
          
            L
          
        
        (
        ϑ
        )
        (
        x
        )
        d
        x
        
          
            =
            by assumption
          
        
        
          ∫
          
            U
          
        
        L
        (
        φ
        )
        (
        x
        )
        ϑ
        (
        x
        )
        d
        x
        =:
        
          
            
              T
            
          
          
            L
            (
            φ
            )
          
        
        (
        ϑ
        )
      
    
    {\displaystyle \Lambda ({\mathcal {T}}_{\varphi })(\vartheta ):=({\mathcal {T}}_{\varphi }\circ {\mathcal {L}})(\vartheta ):=\int _{O}\varphi (x){\mathcal {L}}(\vartheta )(x)dx{\overset {\text{by assumption}}{=}}\int _{U}L(\varphi )(x)\vartheta (x)dx=:{\mathcal {T}}_{L(\varphi )}(\vartheta )}
  Since equality of two functions is equivalent to equality of these two functions evaluated at every point, this shows the desired property.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
We also have a similar lemma for Schwartz distributions:

The proof is exactly word-for-word the same as the one for lemma 4.20.
Noting that multiplication, differentiation and convolution are linear, we will define these operations for distributions by taking 
  
    
      
        L
      
    
    {\displaystyle L}
   in the two above lemmas as the respective of these three operations.

Proof:
The product of two 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
      
    
    {\displaystyle {\mathcal {C}}^{\infty }}
   functions is again 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
      
    
    {\displaystyle {\mathcal {C}}^{\infty }}
  , and further, if 
  
    
      
        φ
        (
        x
        )
        =
        0
      
    
    {\displaystyle \varphi (x)=0}
  , then also 
  
    
      
        (
        f
        φ
        )
        (
        x
        )
        =
        f
        (
        x
        )
        φ
        (
        x
        )
        =
        0
      
    
    {\displaystyle (f\varphi )(x)=f(x)\varphi (x)=0}
  . Hence, 
  
    
      
        f
        φ
        ∈
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle f\varphi \in {\mathcal {D}}(O)}
  .
Also, if 
  
    
      
        
          φ
          
            l
          
        
        →
        φ
      
    
    {\displaystyle \varphi _{l}\to \varphi }
   in the sense of bump functions, then, if 
  
    
      
        K
        ⊂
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle K\subset \mathbb {R} ^{d}}
   is a compact set such that 
  
    
      
        
          supp 
        
        
          φ
          
            n
          
        
        ⊆
        K
      
    
    {\displaystyle {\text{supp }}\varphi _{n}\subseteq K}
   for all 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  ,

  
    
      
        
          
            
              
                ‖
                
                  ∂
                  
                    α
                  
                
                (
                f
                (
                
                  φ
                  
                    l
                  
                
                −
                φ
                )
                )
                
                  ‖
                  
                    ∞
                  
                
              
              
                
                =
                
                  
                    ‖
                    
                      
                        ∑
                        
                          ς
                          ≤
                          α
                        
                      
                      
                        
                          
                            (
                          
                          
                            α
                            ς
                          
                          
                            )
                          
                        
                      
                      
                        ∂
                        
                          ς
                        
                      
                      f
                      
                        ∂
                        
                          α
                          −
                          ς
                        
                      
                      (
                      
                        φ
                        
                          l
                        
                      
                      −
                      φ
                      )
                    
                    ‖
                  
                  
                    ∞
                  
                
              
            
            
              
              
                
                ≤
                
                  ∑
                  
                    ς
                    ≤
                    α
                  
                
                ‖
                
                  ∂
                  
                    ς
                  
                
                f
                
                  ∂
                  
                    α
                    −
                    ς
                  
                
                (
                
                  φ
                  
                    l
                  
                
                −
                φ
                )
                
                  ‖
                  
                    ∞
                  
                
              
            
            
              
              
                
                ≤
                
                  ∑
                  
                    ς
                    ≤
                    α
                  
                
                
                  max
                  
                    x
                    ∈
                    K
                  
                
                
                  |
                
                
                  ∂
                  
                    ς
                  
                
                f
                
                  |
                
                ‖
                
                  ∂
                  
                    α
                    −
                    ς
                  
                
                (
                
                  φ
                  
                    l
                  
                
                −
                φ
                )
                
                  ‖
                  
                    ∞
                  
                
                →
                0
                ,
                l
                →
                ∞
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\|\partial _{\alpha }(f(\varphi _{l}-\varphi ))\|_{\infty }&=\left\|\sum _{\varsigma \leq \alpha }{\binom {\alpha }{\varsigma }}\partial _{\varsigma }f\partial _{\alpha -\varsigma }(\varphi _{l}-\varphi )\right\|_{\infty }\\&\leq \sum _{\varsigma \leq \alpha }\|\partial _{\varsigma }f\partial _{\alpha -\varsigma }(\varphi _{l}-\varphi )\|_{\infty }\\&\leq \sum _{\varsigma \leq \alpha }\max _{x\in K}|\partial _{\varsigma }f|\|\partial _{\alpha -\varsigma }(\varphi _{l}-\varphi )\|_{\infty }\to 0,l\to \infty \end{aligned}}}
  .Hence, 
  
    
      
        f
        
          φ
          
            l
          
        
        →
        f
        φ
      
    
    {\displaystyle f\varphi _{l}\to f\varphi }
   in the sense of bump functions.
Further, also 
  
    
      
        f
        ϕ
        ∈
        
          
            
              C
            
          
          
            ∞
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle f\phi \in {\mathcal {C}}^{\infty }(\mathbb {R} ^{d})}
  . Let 
  
    
      
        α
        ,
        β
        ∈
        
          
            N
          
          
            0
          
          
            d
          
        
      
    
    {\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{d}}
   be arbitrary. Then

  
    
      
        
          ∂
          
            β
          
        
        f
        ϕ
        =
        
          ∑
          
            ς
            ≤
            β
          
        
        
          
            
              (
            
            
              β
              ς
            
            
              )
            
          
        
        
          ∂
          
            ς
          
        
        f
        
          ∂
          
            β
            −
            ς
          
        
        ϕ
      
    
    {\displaystyle \partial _{\beta }f\phi =\sum _{\varsigma \leq \beta }{\binom {\beta }{\varsigma }}\partial _{\varsigma }f\partial _{\beta -\varsigma }\phi }
  .Since all the derivatives of 
  
    
      
        f
      
    
    {\displaystyle f}
   are bounded by polynomials, by the definition of that we obtain

  
    
      
        ∀
        x
        ∈
        
          
            R
          
          
            d
          
        
        :
        
          |
        
        
          ∂
          
            ς
          
        
        f
        (
        x
        )
        
          |
        
        ≤
        
          |
        
        
          p
          
            ς
          
        
        (
        x
        )
        
          |
        
      
    
    {\displaystyle \forall x\in \mathbb {R} ^{d}:|\partial _{\varsigma }f(x)|\leq |p_{\varsigma }(x)|}
  , where 
  
    
      
        
          p
          
            ς
          
        
        ,
        ς
        ∈
        
          
            N
          
          
            0
          
          
            d
          
        
      
    
    {\displaystyle p_{\varsigma },\varsigma \in \mathbb {N} _{0}^{d}}
   are polynomials. Hence,

  
    
      
        ‖
        
          x
          
            α
          
        
        
          ∂
          
            β
          
        
        f
        ϕ
        
          ‖
          
            ∞
          
        
        ≤
        
          ∑
          
            ς
            ≤
            β
          
        
        ‖
        
          x
          
            α
          
        
        
          p
          
            ς
          
        
        
          ∂
          
            β
            −
            ς
          
        
        ϕ
        
          ‖
          
            ∞
          
        
        <
        ∞
      
    
    {\displaystyle \|x^{\alpha }\partial _{\beta }f\phi \|_{\infty }\leq \sum _{\varsigma \leq \beta }\|x^{\alpha }p_{\varsigma }\partial _{\beta -\varsigma }\phi \|_{\infty }<\infty }
  .Similarly, if 
  
    
      
        
          ϕ
          
            l
          
        
        →
        ϕ
      
    
    {\displaystyle \phi _{l}\to \phi }
   in the sense of Schwartz functions, then by exercise 3.6

  
    
      
        ‖
        
          x
          
            α
          
        
        
          ∂
          
            β
          
        
        f
        (
        ϕ
        −
        
          ϕ
          
            l
          
        
        )
        
          ‖
          
            ∞
          
        
        ≤
        
          ∑
          
            ς
            ≤
            β
          
        
        ‖
        
          x
          
            α
          
        
        
          p
          
            ς
          
        
        
          ∂
          
            β
            −
            ς
          
        
        (
        ϕ
        −
        
          ϕ
          
            l
          
        
        )
        
          ‖
          
            ∞
          
        
        →
        0
        ,
        l
        →
        ∞
      
    
    {\displaystyle \|x^{\alpha }\partial _{\beta }f(\phi -\phi _{l})\|_{\infty }\leq \sum _{\varsigma \leq \beta }\|x^{\alpha }p_{\varsigma }\partial _{\beta -\varsigma }(\phi -\phi _{l})\|_{\infty }\to 0,l\to \infty }
  and hence 
  
    
      
        f
        
          ϕ
          
            l
          
        
        →
        f
        ϕ
      
    
    {\displaystyle f\phi _{l}\to f\phi }
   in the sense of Schwartz functions.
If we define 
  
    
      
        L
        (
        φ
        )
        :=
        
          
            L
          
        
        (
        φ
        )
        :=
        f
        φ
      
    
    {\displaystyle L(\varphi ):={\mathcal {L}}(\varphi ):=f\varphi }
  , from lemmas 4.20 and 4.21 follow the other claims.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  

Proof:
We want to apply lemmas 4.20 and 4.21. Hence, we prove that the requirements of these lemmas are met.
Since the derivatives of bump functions are again bump functions, the derivatives of Schwartz functions are again Schwartz functions (see exercise 3.3 for both), and because of theorem 4.22, we have that 
  
    
      
        L
      
    
    {\displaystyle L}
   and 
  
    
      
        
          
            L
          
        
      
    
    {\displaystyle {\mathcal {L}}}
   map 
  
    
      
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}(O)}
   to 
  
    
      
        
          
            D
          
        
        (
        O
        )
      
    
    {\displaystyle {\mathcal {D}}(O)}
  , and if further all 
  
    
      
        
          a
          
            α
          
        
      
    
    {\displaystyle a_{\alpha }}
   and all their derivatives are bounded by polynomials, then 
  
    
      
        L
      
    
    {\displaystyle L}
   and 
  
    
      
        
          
            L
          
        
      
    
    {\displaystyle {\mathcal {L}}}
   map 
  
    
      
        
          
            S
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle {\mathcal {S}}(\mathbb {R} ^{d})}
   to 
  
    
      
        
          
            S
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle {\mathcal {S}}(\mathbb {R} ^{d})}
  .
The sequential continuity of 
  
    
      
        
          
            L
          
        
      
    
    {\displaystyle {\mathcal {L}}}
   follows from theorem 4.22.
Further, for all 
  
    
      
        ϕ
        ,
        θ
        ∈
        
          
            S
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle \phi ,\theta \in {\mathcal {S}}(\mathbb {R} ^{d})}
  ,

  
    
      
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        ϕ
        (
        x
        )
        
          
            L
          
        
        (
        θ
        )
        (
        x
        )
        d
        x
        =
        
          ∑
          
            α
            ∈
            
              
                N
              
              
                0
              
              
                d
              
            
          
        
        (
        −
        1
        
          )
          
            
              |
            
            α
            
              |
            
          
        
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        ϕ
        (
        x
        )
        
          ∂
          
            α
          
        
        (
        
          a
          
            α
          
        
        θ
        )
        (
        x
        )
        d
        x
      
    
    {\displaystyle \int _{\mathbb {R} ^{d}}\phi (x){\mathcal {L}}(\theta )(x)dx=\sum _{\alpha \in \mathbb {N} _{0}^{d}}(-1)^{|\alpha |}\int _{\mathbb {R} ^{d}}\phi (x)\partial _{\alpha }(a_{\alpha }\theta )(x)dx}
  .Further, if we single out an 
  
    
      
        α
        ∈
        
          
            N
          
          
            0
          
          
            d
          
        
      
    
    {\displaystyle \alpha \in \mathbb {N} _{0}^{d}}
  , by Fubini's theorem and integration by parts we obtain

  
    
      
        
          
            
              
                
                  ∫
                  
                    
                      
                        R
                      
                      
                        d
                      
                    
                  
                
                ϕ
                (
                x
                )
                
                  ∂
                  
                    α
                  
                
                (
                
                  a
                  
                    α
                  
                
                θ
                )
                (
                x
                )
                d
                x
              
              
                
                =
                
                  ∫
                  
                    
                      
                        R
                      
                      
                        d
                        −
                        1
                      
                    
                  
                
                
                  ∫
                  
                    
                      R
                    
                  
                
                ϕ
                (
                x
                )
                
                  ∂
                  
                    α
                  
                
                (
                
                  a
                  
                    α
                  
                
                θ
                )
                (
                x
                )
                d
                
                  x
                  
                    1
                  
                
                d
                (
                
                  x
                  
                    2
                  
                
                ,
                …
                ,
                
                  x
                  
                    d
                  
                
                )
              
            
            
              
              
                
                =
                
                  ∫
                  
                    
                      
                        R
                      
                      
                        d
                        −
                        1
                      
                    
                  
                
                
                  ∫
                  
                    
                      R
                    
                  
                
                ϕ
                (
                x
                )
                
                  ∂
                  
                    α
                  
                
                (
                
                  a
                  
                    α
                  
                
                θ
                )
                (
                x
                )
                d
                
                  x
                  
                    1
                  
                
                d
                (
                
                  x
                  
                    2
                  
                
                ,
                …
                ,
                
                  x
                  
                    d
                  
                
                )
              
            
            
              
              
                
                =
                
                  ∫
                  
                    
                      
                        R
                      
                      
                        d
                        −
                        1
                      
                    
                  
                
                (
                −
                1
                
                  )
                  
                    
                      α
                      
                        1
                      
                    
                  
                
                
                  ∫
                  
                    
                      R
                    
                  
                
                
                  ∂
                  
                    (
                    
                      α
                      
                        1
                      
                    
                    ,
                    0
                    ,
                    …
                    ,
                    0
                    )
                  
                
                ϕ
                (
                x
                )
                
                  ∂
                  
                    α
                    −
                    (
                    
                      α
                      
                        1
                      
                    
                    ,
                    0
                    ,
                    …
                    ,
                    0
                    )
                  
                
                (
                
                  a
                  
                    α
                  
                
                θ
                )
                (
                x
                )
                d
                
                  x
                  
                    1
                  
                
                d
                (
                
                  x
                  
                    2
                  
                
                ,
                …
                ,
                
                  x
                  
                    d
                  
                
                )
              
            
            
              
              
                
                =
                ⋯
                =
                (
                −
                1
                
                  )
                  
                    
                      |
                    
                    α
                    
                      |
                    
                  
                
                
                  ∫
                  
                    
                      
                        R
                      
                      
                        d
                      
                    
                  
                
                
                  ∂
                  
                    α
                  
                
                ϕ
                (
                x
                )
                
                  a
                  
                    α
                  
                
                (
                x
                )
                θ
                (
                x
                )
                d
                x
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\int _{\mathbb {R} ^{d}}\phi (x)\partial _{\alpha }(a_{\alpha }\theta )(x)dx&=\int _{\mathbb {R} ^{d-1}}\int _{\mathbb {R} }\phi (x)\partial _{\alpha }(a_{\alpha }\theta )(x)dx_{1}d(x_{2},\ldots ,x_{d})\\&=\int _{\mathbb {R} ^{d-1}}\int _{\mathbb {R} }\phi (x)\partial _{\alpha }(a_{\alpha }\theta )(x)dx_{1}d(x_{2},\ldots ,x_{d})\\&=\int _{\mathbb {R} ^{d-1}}(-1)^{\alpha _{1}}\int _{\mathbb {R} }\partial _{(\alpha _{1},0,\ldots ,0)}\phi (x)\partial _{\alpha -(\alpha _{1},0,\ldots ,0)}(a_{\alpha }\theta )(x)dx_{1}d(x_{2},\ldots ,x_{d})\\&=\cdots =(-1)^{|\alpha |}\int _{\mathbb {R} ^{d}}\partial _{\alpha }\phi (x)a_{\alpha }(x)\theta (x)dx\end{aligned}}}
  .Hence,

  
    
      
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        ϕ
        (
        x
        )
        
          
            L
          
        
        (
        θ
        )
        (
        x
        )
        d
        x
        =
        
          ∫
          
            
              
                R
              
              
                d
              
            
          
        
        L
        (
        ϕ
        )
        (
        x
        )
        θ
        (
        x
        )
        d
        x
      
    
    {\displaystyle \int _{\mathbb {R} ^{d}}\phi (x){\mathcal {L}}(\theta )(x)dx=\int _{\mathbb {R} ^{d}}L(\phi )(x)\theta (x)dx}
  and the lemmas are applicable.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  

Proof:
1.
Let 
  
    
      
        x
        ∈
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle x\in \mathbb {R} ^{d}}
   be arbitrary, and let 
  
    
      
        (
        
          x
          
            l
          
        
        
          )
          
            l
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (x_{l})_{l\in \mathbb {N} }}
   be a sequence converging to 
  
    
      
        x
      
    
    {\displaystyle x}
   and let 
  
    
      
        N
        ∈
        
          N
        
      
    
    {\displaystyle N\in \mathbb {N} }
   such that 
  
    
      
        ∀
        n
        ≥
        N
        :
        ‖
        
          x
          
            n
          
        
        −
        x
        ‖
        ≤
        1
      
    
    {\displaystyle \forall n\geq N:\|x_{n}-x\|\leq 1}
  . Then

  
    
      
        K
        :=
        
          
            
              
                ⋃
                
                  n
                  ≥
                  N
                
              
              
                supp 
              
              φ
              (
              
                x
                
                  n
                
              
              −
              ⋅
              )
              ∪
              
                ⋃
                
                  n
                  <
                  N
                
              
              
                supp 
              
              φ
              (
              
                x
                
                  n
                
              
              −
              ⋅
              )
            
            ¯
          
        
      
    
    {\displaystyle K:={\overline {\bigcup _{n\geq N}{\text{supp }}\varphi (x_{n}-\cdot )\cup \bigcup _{n
        0
      
    
    {\displaystyle \beta _{k}>0}
   (this is possible since otherwise 
  
    
      
        β
        =
        
          0
        
      
    
    {\displaystyle \beta =\mathbf {0} }
  ). Further, we define

  
    
      
        
          e
          
            k
          
        
        :=
        (
        0
        ,
        …
        ,
        0
        ,
        
          
            
              1
              ⏞
            
          
          
            k
            
              th place
            
          
        
        ,
        0
        ,
        …
        ,
        0
        )
      
    
    {\displaystyle e_{k}:=(0,\ldots ,0,\overbrace {1} ^{k{\text{th place}}},0,\ldots ,0)}
  .Then 
  
    
      
        
          |
        
        β
        −
        
          e
          
            k
          
        
        
          |
        
        =
        n
      
    
    {\displaystyle |\beta -e_{k}|=n}
  , and hence 
  
    
      
        
          ∂
          
            β
            −
            
              e
              
                k
              
            
          
        
        (
        
          
            T
          
        
        ∗
        φ
        )
        =
        
          
            T
          
        
        ∗
        (
        
          ∂
          
            β
            −
            
              e
              
                k
              
            
          
        
        φ
        )
      
    
    {\displaystyle \partial _{\beta -e_{k}}({\mathcal {T}}*\varphi )={\mathcal {T}}*(\partial _{\beta -e_{k}}\varphi )}
  .
Furthermore, for all 
  
    
      
        ϑ
        ∈
        
          
            D
          
        
        (
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle \vartheta \in {\mathcal {D}}(\mathbb {R} ^{d})}
  ,

  
    
      
        
          lim
          
            λ
            →
            0
          
        
        
          
            
              
                
                  T
                
              
              ∗
              ϑ
              (
              x
              +
              λ
              
                e
                
                  k
                
              
              )
              −
              
                
                  T
                
              
              ∗
              ϑ
              (
              x
              )
            
            λ
          
        
        =
        
          lim
          
            λ
            →
            0
          
        
        
          
            T
          
        
        
          (
          
            
              
                ϑ
                (
                x
                +
                λ
                
                  e
                  
                    k
                  
                
                −
                ⋅
                )
                −
                ϑ
                (
                x
                −
                ⋅
                )
              
              λ
            
          
          )
        
      
    
    {\displaystyle \lim _{\lambda \to 0}{\frac {{\mathcal {T}}*\vartheta (x+\lambda e_{k})-{\mathcal {T}}*\vartheta (x)}{\lambda }}=\lim _{\lambda \to 0}{\mathcal {T}}\left({\frac {\vartheta (x+\lambda e_{k}-\cdot )-\vartheta (x-\cdot )}{\lambda }}\right)}
  .But due to Schwarz' theorem, 
  
    
      
        
          
            
              ϑ
              (
              x
              +
              λ
              
                e
                
                  k
                
              
              −
              ⋅
              )
              −
              ϑ
              (
              x
              −
              ⋅
              )
            
            λ
          
        
        →
        
          ∂
          
            
              x
              
                k
              
            
          
        
        ϑ
        ,
        λ
        →
        0
      
    
    {\displaystyle {\frac {\vartheta (x+\lambda e_{k}-\cdot )-\vartheta (x-\cdot )}{\lambda }}\to \partial _{x_{k}}\vartheta ,\lambda \to 0}
   in the sense of bump functions, and thus

  
    
      
        
          lim
          
            λ
            →
            0
          
        
        
          
            T
          
        
        
          (
          
            
              
                ϑ
                (
                x
                +
                λ
                
                  e
                  
                    k
                  
                
                −
                ⋅
                )
                −
                ϑ
                (
                x
                −
                ⋅
                )
              
              λ
            
          
          )
        
        =
        
          
            T
          
        
        (
        ϑ
        (
        x
        −
        ⋅
        )
        )
      
    
    {\displaystyle \lim _{\lambda \to 0}{\mathcal {T}}\left({\frac {\vartheta (x+\lambda e_{k}-\cdot )-\vartheta (x-\cdot )}{\lambda }}\right)={\mathcal {T}}(\vartheta (x-\cdot ))}
  .Hence, 
  
    
      
        
          ∂
          
            β
          
        
        (
        
          
            T
          
        
        ∗
        φ
        )
        =
        
          ∂
          
            
              e
              
                k
              
            
          
        
        
          
            T
          
        
        ∗
        (
        
          ∂
          
            β
            −
            
              e
              
                k
              
            
          
        
        φ
        )
        =
        
          
            T
          
        
        ∗
        (
        
          ∂
          
            β
          
        
        φ
        )
      
    
    {\displaystyle \partial _{\beta }({\mathcal {T}}*\varphi )=\partial _{e_{k}}{\mathcal {T}}*(\partial _{\beta -e_{k}}\varphi )={\mathcal {T}}*(\partial _{\beta }\varphi )}
  , since 
  
    
      
        
          ∂
          
            β
            −
            
              e
              
                k
              
            
          
        
        φ
      
    
    {\displaystyle \partial _{\beta -e_{k}}\varphi }
   is a bump function (see exercise 3.3).
3.
This follows from 1. and 2., since 
  
    
      
        
          ∂
          
            β
          
        
        φ
      
    
    {\displaystyle \partial _{\beta }\varphi }
   is a bump function for all 
  
    
      
        β
        ∈
        
          
            N
          
          
            0
          
          
            d
          
        
      
    
    {\displaystyle \beta \in \mathbb {N} _{0}^{d}}
   (see exercise 3.3).
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Exercises ==
Let 
  
    
      
        
          
            
              T
            
          
          
            1
          
        
        ,
        …
        ,
        
          
            
              T
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {T}}_{1},\ldots ,{\mathcal {T}}_{n}}
   be (tempered) distributions and let 
  
    
      
        
          c
          
            1
          
        
        ,
        …
        ,
        
          c
          
            n
          
        
        ∈
        
          R
        
      
    
    {\displaystyle c_{1},\ldots ,c_{n}\in \mathbb {R} }
  . Prove that also 
  
    
      
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
          c
          
            j
          
        
        
          
            
              T
            
          
          
            j
          
        
      
    
    {\displaystyle \sum _{j=1}^{n}c_{j}{\mathcal {T}}_{j}}
   is a (tempered) distribution.
Let 
  
    
      
        f
        :
        
          
            R
          
          
            d
          
        
        →
        
          R
        
      
    
    {\displaystyle f:\mathbb {R} ^{d}\to \mathbb {R} }
   be essentially bounded. Prove that 
  
    
      
        
          
            
              T
            
          
          
            f
          
        
      
    
    {\displaystyle {\mathcal {T}}_{f}}
   is a tempered distribution.
Prove that if 
  
    
      
        
          
            Q
          
        
      
    
    {\displaystyle {\mathcal {Q}}}
   is a set of differentiable functions which go from 
  
    
      
        [
        0
        ,
        1
        
          ]
          
            d
          
        
      
    
    {\displaystyle [0,1]^{d}}
   to 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  , such that there exists a 
  
    
      
        c
        ∈
        
          
            R
          
          
            >
            0
          
        
      
    
    {\displaystyle c\in \mathbb {R} _{>0}}
   such that for all 
  
    
      
        g
        ∈
        
          
            Q
          
        
      
    
    {\displaystyle g\in {\mathcal {Q}}}
   it holds 
  
    
      
        ∀
        x
        ∈
        
          
            R
          
          
            d
          
        
        :
        ‖
        ∇
        g
        (
        x
        )
        ‖
        <
        c
      
    
    {\displaystyle \forall x\in \mathbb {R} ^{d}:\|\nabla g(x)\|