Let X {\displaystyle {\mathcal {X}}} be a linear space. A norm is a real-valued function f {\displaystyle f} on X {\displaystyle {\mathcal {X}}} , with the notation ‖ ⋅ ‖ = f ( ⋅ ) {\displaystyle \|\cdot \|=f(\cdot )} , such that (i) ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ {\displaystyle \|x+y\|\leq \|x\|+\|y\|} (w:triangular inequality) (ii) ‖ λ x ‖ = | λ | ‖ x ‖ {\displaystyle \|\lambda x\|=|\lambda |\|x\|} for any scalar λ {\displaystyle \lambda } (iii) ‖ x ‖ = 0 {\displaystyle \|x\|=0} implies x = 0 {\displaystyle x=0} .(ii) implies that ‖ 0 ‖ = 0 {\displaystyle \|0\|=0} . This and (i) then implies 0 = ‖ x − x ‖ ≤ ‖ x ‖ + ‖ − x ‖ = 2 ‖ x ‖ {\displaystyle 0=\|x-x\|\leq \|x\|+\|-x\|=2\|x\|} for all x {\displaystyle x} ; that is, norms are always non-negative. A linear space with a norm is called a normed space. With the metric d ( x , y ) = ‖ x − y ‖ {\displaystyle d(x,y)=\|x-y\|} a normed space is a metric space. Note that (i) implies that: ‖ x ‖ ≤ ‖ x − y ‖ + ‖ y ‖ {\displaystyle \|x\|\leq \|x-y\|+\|y\|} and ‖ y ‖ ≤ ‖ x − y ‖ + ‖ x ‖ {\displaystyle \|y\|\leq \|x-y\|+\|x\|} and so: | ‖ x ‖ − ‖ y ‖ | ≤ ‖ x − y ‖ {\displaystyle |\|x\|-\|y\||\leq \|x-y\|} . (So, the map x ↦ ‖ x ‖ {\displaystyle x\mapsto \|x\|} is continuous; in fact, 1-Lipschitz continuous.) A complete normed space is called a Banach space. While there is seemingly no prototypical example of a Banach space, we still give one example of a Banach space: C ( K ) {\displaystyle {\mathcal {C}}(K)} , the space of all continuous functions on a compact space K {\displaystyle K} , can be identified with a Banach space by introducing the norm: ‖ ⋅ ‖ = sup K | ⋅ | {\displaystyle \|\cdot \|=\sup _{K}|\cdot |} It is a routine exercise to check that this is indeed a norm. The completeness holds since, from real analysis, we know that a uniform limit of a sequence of continuous functions is continuous. In concrete spaces like this one, one can directly show the completeness. More often than that, however, we will see that the completeness is a necessary condition for some results (especially, reflexivity), and thus the space has to be complete. The matter will be picked up in the later chapters. Another example of Banach spaces, which is more historical, is an l p {\displaystyle l_{p}} space; that is, the space of convergent series. (The geometric properties of l p {\displaystyle l_{p}} spaces will be investigated in Chapter 4.) It is clear that l p {\displaystyle l_{p}} is a linear space, since the sum of two p-convergent series is again p-convergent. That the l p {\displaystyle l_{p}} norm is in fact a norm follows from Lemma 2.1. If ‖ x ‖ p < ∞ {\displaystyle \|x\|_{p}<\infty } , then ‖ x ‖ p = sup ‖ y ‖ q = 1 | ∑ x j y j ¯ | {\displaystyle \|x\|_{p}=\sup _{\|y\|_{q}=1}|\sum x_{j}{\overline {y_{j}}}|} , where 1 / p + 1 / q = 1 {\displaystyle 1/p+1/q=1} Proof. By Hölder's inequality, sup ‖ y ‖ q = 1 | ∑ x j y j ¯ | ≤ ‖ x ‖ p {\displaystyle \sup _{\|y\|_{q}=1}|\sum x_{j}{\overline {y_{j}}}|\leq \|x\|_{p}} Conversely, if ‖ x ‖ p = 1 {\displaystyle \|x\|_{p}=1} , then taking y j = x j ¯ | x j | p − 2 {\displaystyle y_{j}={\overline {x_{j}}}|x_{j}|^{p-2}} we have: ∑ x j y j ¯ = ∑ | x j | p = 1 {\displaystyle \sum x_{j}{\overline {y_{j}}}=\sum |x_{j}|^{p}=1} , while ‖ y ‖ q = ( ∑ | x j | ( p − 1 ) q ) 1 / q = 1 {\displaystyle \|y\|_{q}=(\sum |x_{j}|^{(p-1)q})^{1/q}=1} .since p = ( p − 1 ) q {\displaystyle p=(p-1)q} . More generally, if ‖ x ‖ p ≠ 0 {\displaystyle \|x\|_{p}\neq 0} , then x ‖ x ‖ p = sup ‖ y ‖ q = 1 | ∑ x j y j ¯ | 1 ‖ x ‖ p {\displaystyle {x \over \|x\|_{p}}=\sup _{\|y\|_{q}=1}|\sum x_{j}{\overline {y_{j}}}|{1 \over \|x\|_{p}}} .Since the identity is obvious when x = 0 {\displaystyle x=0} , the proof is complete. ◻ {\displaystyle \square } Now, it remains to show that an l p {\displaystyle l_{p}} space is complete. For that, let x k ∈ l p {\displaystyle x_{k}\in l_{p}} be a Cauchy sequence. This means explicitly that ∑ n = 1 ∞ | ( x k ) n − ( x j ) n | 2 → 0 {\displaystyle \sum _{n=1}^{\infty }|(x_{k})_{n}-(x_{j})_{n}|^{2}\to 0} as n , m → ∞ {\displaystyle n,m\to \infty } For each n {\displaystyle n} , by completeness, lim k → ∞ ( x k ) n {\displaystyle \lim _{k\to \infty }(x_{k})_{n}} exists and we denote it by y n {\displaystyle y_{n}} . Let ϵ > 0 {\displaystyle \epsilon >0} be given. Since x k {\displaystyle x_{k}} is Cauchy, there is N {\displaystyle N} such that ∑ n = 1 ∞ | ( x k ) n − ( x j ) n | 2 < ϵ {\displaystyle \sum _{n=1}^{\infty }|(x_{k})_{n}-(x_{j})_{n}|^{2}<\epsilon } for k , j > N {\displaystyle k,j>N} Then, for any k > N {\displaystyle k>N} , ∑ n = 1 ∞ | ( x k ) n − y n | 2 = sup m ≥ 1 ∑ n = 1 m | ( x k ) n − y n | 2 = sup m ≥ 1 lim j → ∞ ∑ n = 1 m | ( x k ) n − ( x j ) n | 2 < ϵ {\displaystyle \sum _{n=1}^{\infty }|(x_{k})_{n}-y_{n}|^{2}=\sup _{m\geq 1}\sum _{n=1}^{m}|(x_{k})_{n}-y_{n}|^{2}=\sup _{m\geq 1}\lim _{j\to \infty }\sum _{n=1}^{m}|(x_{k})_{n}-(x_{j})_{n}|^{2}<\epsilon } Hence, x k → y {\displaystyle x_{k}\to y} with y = ∑ n = 1 ∞ y n {\displaystyle y=\sum _{n=1}^{\infty }y_{n}} . y {\displaystyle y} is in fact in l p {\displaystyle l_{p}} since ‖ y ‖ 2 ≤ ‖ y − x n ‖ 2 + ‖ x n ‖ 2 < ∞ {\displaystyle \|y\|_{2}\leq \|y-x_{n}\|_{2}+\|x_{n}\|_{2}<\infty } . (We stress the fact that the completeness of l p {\displaystyle l_{p}} spaces come from the fact that the field of complex numbers is complete; in other words, l p {\displaystyle l_{p}} spaces may fail to be complete if the base field is not complete.) l p {\displaystyle l_{p}} is also separable; i.e., it has a countable dense subset. This follows from the fact that l p {\displaystyle l_{p}} can be written as a union of subspaces with dimensions 1, 2, ..., which are separable. (TODO: need more details.) We define the operator norm of a continuous linear operator f {\displaystyle f} between normed spaces X {\displaystyle {\mathcal {X}}} and Y {\displaystyle {\mathcal {Y}}} , denoted by ‖ f ‖ {\displaystyle \|f\|} , by ‖ f ‖ = sup ‖ x ‖ X ≤ 1 ‖ f ( x ) ‖ Y {\displaystyle \|f\|=\sup _{\|x\|_{\mathcal {X}}\leq 1}\|f(x)\|_{\mathcal {Y}}} 2 Theorem Let T {\displaystyle T} be a linear operator from a normed space X {\displaystyle {\mathcal {X}}} to a normed space Y {\displaystyle {\mathcal {Y}}} . (i) T {\displaystyle T} is continuous if and only if there is a constant C > 0 {\displaystyle C>0} such that ‖ T x ‖ ≤ C ‖ x ‖ {\displaystyle \|Tx\|\leq C\|x\|} for all x ∈ X {\displaystyle x\in X} (ii) ‖ f ‖ = inf { {\displaystyle \|f\|=\inf\{} any C {\displaystyle C} as in (i) } = sup ‖ x ‖ = 1 ‖ f ( x ) ‖ {\displaystyle \}=\sup _{\|x\|=1}\|f(x)\|} if X {\displaystyle {\mathcal {X}}} has nonzero element. (Recall that the inf of the empty set is ∞ {\displaystyle \infty } .)Proof: If ‖ T ‖ < ∞ {\displaystyle \|T\|<\infty } , then ‖ T ( x n − x ) ‖ Y ≤ ‖ T ‖ ‖ x n − x ‖ X → 0 {\displaystyle \|T(x_{n}-x)\|_{\mathcal {Y}}\leq \|T\|\|x_{n}-x\|_{\mathcal {X}}\to 0} as x n → x {\displaystyle x_{n}\to x} . Hence, T {\displaystyle T} is continuous. Conversely, suppose ‖ T ‖ = ∞ {\displaystyle \|T\|=\infty } . Then we can find x n ∈ X {\displaystyle x_{n}\in {\mathcal {X}}} with ‖ x n ‖ X ≤ 1 {\displaystyle \|x_{n}\|_{\mathcal {X}}\leq 1} and ‖ T x n ‖ ≥ n {\displaystyle \|Tx_{n}\|\geq n} . Then x n n → 0 {\displaystyle {x_{n} \over n}\to 0} while ‖ T ( x n n ) ‖ ↛ 0 {\displaystyle \left\|T\left({x_{n} \over n}\right)\right\|\not \to 0} . Hence, T {\displaystyle T} is not continuous. The proof of (i) is complete. For (ii), see w:operator norm for now. (TODO: write an actual proof). ◻ {\displaystyle \square } It is clear that an addition and a scalar multiplication are both continuous. (Use a sequence to check this.) Since the inverse of an addition is again addition, an addition is also an open mapping. Ditto to nonzero-scalar multiplications. In other words, translations and dilations of open (resp. closed) sets are again open (resp. closed). Not all linear operators are continuous. Take the linear operator defined by D ( P ) = X P ′ {\displaystyle D(P)=XP'} on the normed vector space of polynomials R [ X ] {\displaystyle \mathbb {R} [X]} with the suprenum norm ‖ P ‖ ∞ = sup x ∈ [ 0 , 1 ] | P ( x ) | {\displaystyle \|P\|_{\infty }=\sup _{x\in [0,1]}|P(x)|} ; since D ( X n ) = n X n {\displaystyle D(X^{n})=nX^{n}} , the unit ball is not bounded and hence this linear operator is not continuous. Notice that the kernel of this non continuous linear operator is closed: ker D = { 0 } {\displaystyle \ker D=\{0\}} . However, when a linear operator is of finite rank, the closeness of the kernel is in fact synonymous to continuity. To see this, we start with the special case of linear forms. 2 Theorem A (non null) linear form is continuous iff it's kernel is closed. T {\displaystyle T} continuous ⇔ ker T = ker T ¯ {\displaystyle \Leftrightarrow \ker T={\overline {\ker T}}} Proof: If the linear form T {\displaystyle T} on a normed vector space X {\displaystyle X} is continuous, then it's kernel is closed since it's the continuous inverse image of the closed set { 0 } {\displaystyle \{0\}} . Conversely, suppose a linear form T : X → R {\displaystyle T:X\to \mathbb {R} } is not continuous. then by the previous theorem, ∀ c > 0 , ∃ x ( c ) s . t . | T x ( c ) | ⩾ c ‖ x ( c ) ‖ {\displaystyle \forall c>0,\exists x(c)s.t.|Tx(c)|\geqslant c\|x(c)\|} so in particular, one can define a sequence { x n } {\displaystyle \{x_{n}\}} such that | T x n | ⩾ n ‖ x n ‖ > 0 {\displaystyle |Tx_{n}|\geqslant n\|x_{n}\|>0} . Then denote: u n := x n | x n | {\displaystyle u_{n}:={\frac {x_{n}}{|x_{n}|}}} , one has defined a unit normed sequence ( | u n | = 1 {\displaystyle |u_{n}|=1} ) s.t. | T u n | ⩾ n {\displaystyle |Tu_{n}|\geqslant n} . Furthermore, denote v n := u n | T u n | {\displaystyle v_{n}:={\dfrac {u_{n}}{|Tu_{n}|}}} . Since | u n | | T u n | ⩽ 1 n {\displaystyle {\frac {|u_{n}|}{|Tu_{n}|}}\leqslant {\dfrac {1}{n}}} , one can define a sequence that converges { v n } → 0 {\displaystyle \{v_{n}\}\to 0} whilst | T v n | = 1 {\displaystyle |Tv_{n}|=1} .Now, since ker T ≠ X {\displaystyle \ker T\neq X} , then there exists a {\displaystyle a} such that T a ≠ 0 {\displaystyle Ta\neq 0} . Then the sequence of general term converges a − v n T a ⏟ ∈ ker T → a ∉ ker T {\displaystyle \underbrace {a-v_{n}Ta} _{\in \ker T}\to a\notin \ker T} and hence ker T {\displaystyle \ker T} is not closed. ◻ {\displaystyle \square } Furthermore, if the linear form is continuous and the kernel is dense, then ker T = ker T ¯ = X ⇒ f = 0 {\displaystyle \ker T={\overline {\ker T}}=X\Rightarrow f=0} , hence a continuous & non null linear has a non dense kernel, and hence a linear form with a dense kernel is whether null or non continuous so a non null continuous linear form with a dense kernel is not continuous, and a linear form with a dense kernel is not continuous. 2 Corollary' " A non null linear form on a normed vector space is not continuous iff it's kernel is dense. ker T ¯ = X ⇔ T {\displaystyle {\overline {\ker T}}=X\Leftrightarrow T} is not continuous"More generally, we have: 2 Theorem " A non null linear operator of finite rank between normed vector spaces. then closeness of the null space is equivalent to continuity." Proof: It remains to show that continuity implies closeness of the kernel. Suppose T : X → Y {\displaystyle T:X\to Y} is not continuous. Denote r := dim Im T {\displaystyle r:=\dim {\mbox{Im }}T} ; 2 Lemma If T : X → Y {\displaystyle T:X\to Y} is a linear operator between normed vector spaces, then T {\displaystyle T} is of finite rank r {\displaystyle r} iff there exists r {\displaystyle r} independent linear forms ( f 1 , … , f r ) {\displaystyle (f_{1},\dots ,f_{r})} and r {\displaystyle r} independent vectors ( a 1 , … , a r ) {\displaystyle (a_{1},\dots ,a_{r})} such that T x = a 1 f 1 ( x ) + ⋯ + a r f r ( x ) {\displaystyle Tx=a_{1}f_{1}(x)+\dots +a_{r}f_{r}(x)} " Proof: take a basis ( a 1 , … , a r ) {\displaystyle (a_{1},\dots ,a_{r})} of Im T {\displaystyle {\mbox{Im }}T} , then from T x = ∑ i = 1 r a i y i {\displaystyle Tx=\sum _{i=1}^{r}a_{i}y_{i}} , one can define r {\displaystyle r} mappings f i ( x ) = y i {\displaystyle f_{i}(x)=y_{i}} . Unicity and linearity of T {\displaystyle T} implies linearity of the f i {\displaystyle f_{i}} 's. Furthermore, the family ( f 1 , … , f r ) {\displaystyle (f_{1},\dots ,f_{r})} of linear forms of X ∗ {\displaystyle X^{*}} is linearly independent: suppose not, then there exist a non zero family ( α 1 , … , α r ) {\displaystyle (\alpha _{1},\dots ,\alpha _{r})} such that e.g. f 1 = ∑ i = 2 r α i f i {\displaystyle f_{1}=\sum _{i=2}^{r}\alpha _{i}f_{i}} so T x = ∑ i = 1 r f i ( x ) a i = ( ∑ i = 2 r α i f i ( x ) ) a 1 + ∑ i = 2 r f i ( x ) a i = ( α 2 a 1 + a 2 ) f 2 ( x ) + ⋯ + ( α n a 1 + a r ) f r ( x ) {\displaystyle Tx=\sum _{i=1}^{r}f_{i}(x)a_{i}=(\sum _{i=2}^{r}\alpha _{i}f_{i}(x))a_{1}+\sum _{i=2}^{r}f_{i}(x)a_{i}=(\alpha _{2}a_{1}+a_{2})f_{2}(x)+\dots +(\alpha _{n}a_{1}+a_{r})f_{r}(x)} and the family ( α i a 1 + a i ) i = 2 , … , r {\displaystyle (\alpha _{i}a_{1}+a_{i})_{i=2,\dots ,r}} spans Im T {\displaystyle {\mbox{Im }}T} , so dim Im T = r − 1 {\displaystyle \dim {\mbox{Im }}T=r-1} which is a contradiction. Finally, one has a unique decomposition of a finite rank linear operator: T x = a 1 f 1 ( x ) + … a r f r ( x ) {\displaystyle Tx=a_{1}f_{1}(x)+\dots a_{r}f_{r}(x)} with f i ∈ E ∗ {\displaystyle f_{i}\in E^{*}} ◻ {\displaystyle \square } Take x ∈ ker T ⇒ x ∈ ⋂ i = 1 r ker f i ⊂ ker f i {\displaystyle x\in \ker T\Rightarrow x\in \bigcap _{i=1}^{r}\ker f_{i}\subset \ker f_{i}} . Then there exists a vector subspace H i {\displaystyle H_{i}} such that ker f i = ker T ⊕ H i {\displaystyle \ker f_{i}=\ker T\oplus H_{i}} . Denote T i : H i → Im T {\displaystyle T_{i}:H_{i}\to {\mbox{Im}}T} the restriction of T {\displaystyle T} to H i {\displaystyle H_{i}} . Since k e r T i = { x ∈ H i : T i ( x ) } = 0 = ker T ∩ H i = { 0 } {\displaystyle kerT_{i}=\{x\in H_{i}:T_{i}(x)\}=0=\ker T\cap H_{i}=\{0\}} , the linear operator T i {\displaystyle T_{i}} is injective so Im T i ⊂ Im T {\displaystyle {\mbox{Im}}T_{i}\subset {\mbox{Im}}T} and H i {\displaystyle H_{i}} is of finite dimension, and this for all i = 1 , … , r {\displaystyle i=1,\dots ,r} . By hypothesis ker T {\displaystyle \ker T} is closed. Since the sum of this closed subspace and a subspace of finite dimension ( H i {\displaystyle H_{i}} ) is closed (see lemma bellow), it follows that the kernel of each r {\displaystyle r} linear forms ker f i {\displaystyle \ker f_{i}} is closed, so the f i {\displaystyle f_{i}} 's are all continuous by the first case and hence T {\displaystyle T} is continuous. ◻ {\displaystyle \square } 2 Lemma The sum of subspace of finite dimension with a closed subspace is closed." Proof: by induction on the dimension. Case n = 1 {\displaystyle n=1} . Let's show that H := F + K a {\displaystyle H:=F+\mathbb {K} a} is closed when F {\displaystyle F} is closed (where K {\displaystyle \mathbb {K} } is a complete field). Any x ∈ H {\displaystyle x\in H} can be uniquely written as x = y + λ a {\displaystyle x=y+\lambda a} with y ∈ F {\displaystyle y\in F} . There exists a linear form L {\displaystyle L} s.t. x = y + L ( x ) a {\displaystyle x=y+L(x)a} . Since L {\displaystyle L} is closed in ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} so in ( H , ‖ ⋅ ‖ ) {\displaystyle (H,\|\cdot \|)} , then f {\displaystyle f} is continuous by the first case. Take a convergente sequence x n → x ∈ E {\displaystyle {x_{n}}\to x\in E} of H {\displaystyle H} . He have x n = y n + L ( x n ) a {\displaystyle x_{n}=y_{n}+L(x_{n})a} with y n ∈ F {\displaystyle y_{n}\in F} . Since the sequence x n {\displaystyle {x_{n}}} is convergente, then it si Cauchy, so it's continuous image L ( x n ) {\displaystyle {L(x_{n})}} is also Cauchy. Since R {\displaystyle \mathbb {R} } is complete, then L ( x n ) → λ {\displaystyle {L(x_{n})}\to \lambda } . Finally, the sequence y n {\displaystyle {y_{n}}} converges to x − λ a {\displaystyle x-\lambda a} . Since F {\displaystyle F} is closed, then x − λ a ∈ F {\displaystyle x-\lambda a\in F} and x ∈ H {\displaystyle x\in H} so H {\displaystyle H} is closed. Suppose the result holds for all subspaces of dimension ⩽ p {\displaystyle \leqslant p} . Let G {\displaystyle G} be a subspace of dimension p + 1 {\displaystyle p+1} . Let ( a 1 , … , a p + 1 ) {\displaystyle (a_{1},\dots ,a_{p+1})} be a basis of G {\displaystyle G} . Then H := F + ⨁ i = 1 p K a i + K a p + 1 {\displaystyle H:=F+\bigoplus _{i=1}^{p}\mathbb {K} a_{i}+\mathbb {K} a_{p+1}} and concludes easily. ◻ {\displaystyle \square } 2 Corollary Any linear operator on a normed vector space of finite dimension onto a normed vector space is continuous. Proof: Since X {\displaystyle X} is of finite dimension, then any linear operator is of finite rank. Then as dim ker T + dim Im T = dim X {\displaystyle \dim \ker T+\dim {\mbox{Im}}T=\dim X} holds, it comes that the null space is of finite dimension, so is closed (any K {\displaystyle \mathbb {K} } vector subspace of finite dimension n {\displaystyle n} is isomorphic to K n {\displaystyle \mathbb {K} ^{n}} (where K {\displaystyle \mathbb {K} } is a complete field), so the subspace is complete and closed). Then one applies the previous theorem. ◻ {\displaystyle \square } 2 Lemma (Riesz) A normed space X {\displaystyle {\mathcal {X}}} is finite-dimensional if and only if its closed unit ball is compact. Proof: Let T : C n → X {\displaystyle T:\mathbf {C} ^{n}\to X} be a linear vector space isomorphism. Since T {\displaystyle T} has closed kernel, arguing as in the proof of the preceding theorem, we see that T {\displaystyle T} is continuous. By the same reasoning T − 1 {\displaystyle T^{-1}} is continuous. It follows: { x ∈ X | ‖ x ‖ ≤ 1 } ⊂ T { y ∈ C n | ‖ y ‖ ≤ ‖ T − 1 ‖ } {\displaystyle \{x\in {\mathcal {X}}|\|x\|\leq 1\}\subset T\{y\in \mathbf {C} ^{n}|\|y\|\leq \|T^{-1}\|\}} In the above, the left-hand side is closed, and the right-hand is a continuous image of a closed ball, which is compact. Hence, the closed unit ball is a subset of a compact set and thus compact. Now, the converse. If X {\displaystyle {\mathcal {X}}} is not finite dimensional, we can construct a sequence x j {\displaystyle x_{j}} such that: 1 = ‖ x j ‖ ≤ ‖ x j − ∑ k = 1 j − 1 a k x k ‖ {\displaystyle 1=\|x_{j}\|\leq \|x_{j}-\sum _{k=1}^{j-1}a_{k}x_{k}\|} for any sequence of scalars a k {\displaystyle a_{k}} .Thus, in particular, ‖ x j − x k ‖ ≥ 1 {\displaystyle \|x_{j}-x_{k}\|\geq 1} for all j , k {\displaystyle j,k} . (For the details of this argument, see : w:Riesz's lemma for now) ◻ {\displaystyle \square } 2 Corollary Every finite-dimensional normed space is a Banach space. Proof: Let x n {\displaystyle x_{n}} be a Cauchy sequence. Since it is bounded, it is contained in some closed ball, which is compact. x n {\displaystyle x_{n}} thus has a convergent subsequence and so x n {\displaystyle x_{n}} itself converges. ◻ {\displaystyle \square } 2 Theorem A normed space X {\displaystyle {\mathcal {X}}} is finite-dimensional if and only if every linear operator T {\displaystyle T} defined on X {\displaystyle {\mathcal {X}}} is continuous. Proof: Identifying the range of T {\displaystyle T} with C n {\displaystyle \mathbf {C} ^{n}} , we can write: T x = ( f 1 ( x ) , f 2 ( x ) , . . . f n ( x ) ) {\displaystyle Tx=(f_{1}(x),f_{2}(x),...f_{n}(x))} where f 1 , . . . f n {\displaystyle f_{1},...f_{n}} are linear functionals. The dimensions of the kernels of f j {\displaystyle f_{j}} are finite. Thus, f j {\displaystyle f_{j}} all have complete and thus closed kernels. Hence, they are continuous and so T {\displaystyle T} is continuous. For the converse, we need Axiom of Choice. (TODO: complete the proof.) ◻ {\displaystyle \square } The graph of any function f {\displaystyle f} on a set E {\displaystyle E} is the set { ( x , f ( x ) ) | x ∈ E } {\displaystyle \{(x,f(x))|x\in E\}} . A continuous function between metric spaces has closed graph. In fact, suppose ( x j , f ( x j ) ) → ( x , y ) {\displaystyle (x_{j},f(x_{j}))\to (x,y)} . By continuity, f ( x j ) → f ( x ) {\displaystyle f(x_{j})\to f(x)} ; in other words, y = f ( x ) {\displaystyle y=f(x)} and so ( x , y ) {\displaystyle (x,y)} is in the graph of f {\displaystyle f} . It follows (in the next theorem) that a continuous linear operator with closed graph has closed domain. (Note the continuity here is a key; we will shortly study a linear operator that has closed graph but has non-closed domain.) 2 Theorem Let T : X → Y {\displaystyle T:{\mathcal {X}}\to {\mathcal {Y}}} be a continuous densely defined linear operator between Banach spaces. Then its domain is closed; i.e., T {\displaystyle T} is actually defined everywhere. Proof: Suppose f j → f {\displaystyle f_{j}\to f} and T f j {\displaystyle Tf_{j}} is defined for every j {\displaystyle j} ; i.e., the sequence f j {\displaystyle f_{j}} is in the domain of T {\displaystyle T} . Since ‖ T f j − T f k ‖ ≤ ‖ T ‖ ‖ f j − f k ‖ → 0 {\displaystyle \|Tf_{j}-Tf_{k}\|\leq \|T\|\|f_{j}-f_{k}\|\to 0} , T f j {\displaystyle Tf_{j}} is Cauchy. It follows that ( f j , T f j ) {\displaystyle (f_{j},Tf_{j})} is Cauchy and, by completeness, has limit ( g , T g ) {\displaystyle (g,Tg)} since the graph of T is closed. Since f = g {\displaystyle f=g} , T f {\displaystyle Tf} is defined; i.e., f {\displaystyle f} is in the domain of T {\displaystyle T} . ◻ {\displaystyle \square } The theorem is frequently useful in application. Suppose we wish to prove some linear formula. We first show it holds for a function with compact support and of varying smoothness, which is usually easy to do because the function vanishes on the boundary, where much of complications reside. Because of th linear nature in the formula, the theorem then tells that the formula is true for the space where the above functions are dense. We shall now turn our attention to the consequences of the fact that a complete metric space is a Baire space. They tend to be more significant than results obtained by directly appealing to the completeness. Note that not every normed space that is a Baire space is a Banach space. 2 Theorem (open mapping theorem) Let X , Y {\displaystyle {\mathcal {X}},{\mathcal {Y}}} be Banach spaces. If T : X → Y {\displaystyle T:{\mathcal {X}}\to {\mathcal {Y}}} is a continuous linear surjection, then it is an open mapping; i.e., it maps open sets to open sets. Proof: Let B ( r ) = { x ∈ X ; ‖ x ‖ < r } {\displaystyle B(r)=\{x\in {\mathcal {X}};\|x\|0 {\displaystyle K>0} such that if y ∈ im ( T ) {\displaystyle y\in \operatorname {im} (T)} then ‖ x ‖ ≤ K ‖ y ‖ {\displaystyle \|x\|\leq K\|y\|} for some x {\displaystyle x} with T x = y {\displaystyle Tx=y} . Proof: This is immediate once we have the notion of a quotient map, which we now define as follows. Let M {\displaystyle {\mathcal {M}}} be a closed subspace of a normed space X {\displaystyle {\mathcal {X}}} . The quotient space X / M {\displaystyle {\mathcal {X}}/{\mathcal {M}}} is a normed space with norm: ‖ π ( x ) ‖ = inf { ‖ x + m ‖ ; m ∈ M } {\displaystyle \|\pi (x)\|=\inf\{\|x+m\|;m\in {\mathcal {M}}\}} where π : X → X / M {\displaystyle \pi :{\mathcal {X}}\to {\mathcal {X}}/{\mathcal {M}}} is a canonical projection. That ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is a norm is obvious except for the triangular inequality. But since ‖ π ( x + y ) ‖ ≤ ‖ x + m ‖ + ‖ y + n ‖ {\displaystyle \|\pi (x+y)\|\leq \|x+m\|+\|y+n\|} for all m , n ∈ M {\displaystyle m,n\in {\mathcal {M}}} . Taking inf over m , n {\displaystyle m,n} separately we get: ‖ π ( x + y ) ‖ ≤ ‖ π ( x ) ‖ + ‖ π ( y ) ‖ {\displaystyle \|\pi (x+y)\|\leq \|\pi (x)\|+\|\pi (y)\|} Suppose, further, that X {\displaystyle {\mathcal {X}}} is also a commutative algebra and M {\displaystyle {\mathcal {M}}} is an ideal. Then X / M {\displaystyle {\mathcal {X}}/{\mathcal {M}}} becomes a quotient algebra. In fact, as above, we have: ‖ π ( x ) π ( y ) ‖ = ‖ π ( ( x + m ) ( y + n ) ) ‖ ≤ ‖ x + m ‖ ‖ y + n ‖ {\displaystyle \|\pi (x)\pi (y)\|=\|\pi ((x+m)(y+n))\|\leq \|x+m\|\|y+n\|} ,for all m , n ∈ M {\displaystyle m,n\in {\mathcal {M}}} since π {\displaystyle \pi } is a homomorphism. Taking inf completes the proof. So, the only nontrivial question is the completeness. It turns out that X / M {\displaystyle {\mathcal {X}}/{\mathcal {M}}} is a Banach space (or algebra) if X {\displaystyle {\mathcal {X}}} is Banach space (or algebra). In fact, suppose ∑ n = 1 ∞ ‖ π ( x n ) ‖ < ∞ {\displaystyle \sum _{n=1}^{\infty }\|\pi (x_{n})\|<\infty } Then we can find a sequence y n ∈ M {\displaystyle y_{n}\in {\mathcal {M}}} such that ∑ n = 1 ∞ ‖ x n + y n ‖ < ∞ {\displaystyle \sum _{n=1}^{\infty }\|x_{n}+y_{n}\|<\infty } By completeness, ∑ n = 1 ∞ x n + y n {\displaystyle \sum _{n=1}^{\infty }x_{n}+y_{n}} converges, and since π {\displaystyle \pi } is continuous, ∑ n = 1 ∞ π ( x n ) {\displaystyle \sum _{n=1}^{\infty }\pi (x_{n})} converges then. The completeness now follows from: 2 Lemma Let X {\displaystyle {\mathcal {X}}} be a normed space. Then X {\displaystyle {\mathcal {X}}} is complete (thus a Banach space) if and only if ∑ n = 1 ∞ ‖ x n ‖ < ∞ {\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|<\infty } implies ∑ n = 1 ∞ x n {\displaystyle \sum _{n=1}^{\infty }x_{n}} converges.Proof: ( ⇒ {\displaystyle \Rightarrow } ) We have: ‖ ∑ n = k k + m x n ‖ ≤ ∑ n = k k + m ‖ x n ‖ {\displaystyle \|\sum _{n=k}^{k+m}x_{n}\|\leq \sum _{n=k}^{k+m}\|x_{n}\|} .By hypothesis, the right-hand side goes to 0 as n , m → ∞ {\displaystyle n,m\to \infty } . By completeness, ∑ n = 1 ∞ x n {\displaystyle \sum _{n=1}^{\infty }x_{n}} converges. Conversely, suppose x j {\displaystyle x_{j}} is a Cauchy sequence. Thus, for each j = 1 , 2 , . . . {\displaystyle j=1,2,...} , there exists an index k j {\displaystyle k_{j}} such that ‖ x n − x m ‖ < 2 − j {\displaystyle \|x_{n}-x_{m}\|<2^{-j}} for any n , m ≥ k j {\displaystyle n,m\geq k_{j}} . Let x k 0 = 0 {\displaystyle x_{k_{0}}=0} . Then ∑ j = 0 ∞ ‖ x k j + 1 − x k j ‖ < 2 {\displaystyle \sum _{j=0}^{\infty }\|x_{k_{j+1}}-x_{k_{j}}\|<2} . Hence, by assumption we can get the limit x = ∑ j = 0 ∞ x k j + 1 − x k j {\displaystyle x=\sum _{j=0}^{\infty }x_{k_{j+1}}-x_{k_{j}}} , and since ‖ x n k − x ‖ = ‖ ∑ j = 1 n x k j + 1 − x k j − x ‖ → 0 {\displaystyle \|x_{n_{k}}-x\|=\|\sum _{j=1}^{n}x_{k_{j+1}}-x_{k_{j}}-x\|\to 0} as n → ∞ {\displaystyle n\to \infty } ,we conclude that x j {\displaystyle x_{j}} has a subsequence converging to x {\displaystyle x} ; thus, it converges to x {\displaystyle x} . ◻ {\displaystyle \square } The next result is arguably the most important theorem in the theory of Banach spaces. (At least, it is used the most frequently in application.) 2 Theorem (closed graph theorem) Let X , Y {\displaystyle {\mathcal {X}},{\mathcal {Y}}} be Banach spaces, and T : X → Y {\displaystyle T:{\mathcal {X}}\to {\mathcal {Y}}} a linear operator. The following are equivalent. (i) T {\displaystyle T} is continuous. (ii) If x j → 0 {\displaystyle x_{j}\to 0} and T x j {\displaystyle Tx_{j}} is convergent, then T x j → 0 {\displaystyle Tx_{j}\to 0} . (iii) The graph of T {\displaystyle T} is closed.Proof: That (i) implies (ii) is clear. To show (iii), suppose ( x j , T x j ) {\displaystyle (x_{j},Tx_{j})} is convergent in X {\displaystyle X} . Then x j {\displaystyle x_{j}} converges to some x 0 {\displaystyle x_{0}} or x j − x 0 → 0 {\displaystyle x_{j}-x_{0}\to 0} , and T x j − T x {\displaystyle Tx_{j}-Tx} is convergent. Thus, if (ii) holds, T ( x j − x ) → 0 {\displaystyle T(x_{j}-x)\to 0} . Finally, to prove (iii) ⇒ {\displaystyle \Rightarrow } (i), we note that Corollary 2.something gives the inequality: ‖ ⋅ ‖ + ‖ T ⋅ ‖ ≤ K ‖ ⋅ ‖ {\displaystyle \|\cdot \|+\|T\cdot \|\leq K\|\cdot \|} since by hypothesis the norm in the left-hand side is complete. Hence, if x j → x {\displaystyle x_{j}\to x} , then T x j → T x {\displaystyle Tx_{j}\to Tx} . ◻ {\displaystyle \square } Note that when the domain of a linear operator is not a Banach space (e.g., just dense in a Banach space), the condition (ii) is not sufficient for the graph of the operator to be closed. (It is not hard to find an example of this in other fields, but the reader might want to construct one himself as an exercise.) Finally, note that an injective linear operator has closed graph if and only if its inverse is closed, since the map ( x 1 , x 2 ) ↦ x 2 , x 1 {\displaystyle (x_{1},x_{2})\mapsto x_{2},x_{1}} sends closed sets to closed sets. 2 Theorem Let ( X j , ‖ ⋅ ‖ j ) {\displaystyle ({\mathcal {X}}_{j},\|\cdot \|_{j})} be Banach spaces. Let T : X 1 → X 2 {\displaystyle T:{\mathcal {X}}_{1}\to {\mathcal {X}}_{2}} be a closed densely defined operator and S {\displaystyle S} be a linear operator with dom ( T ) ⊂ dom ( S ) {\displaystyle \operatorname {dom} (T)\subset \operatorname {dom} (S)} . If there are constants a , b {\displaystyle a,b} such that (i) 0 ≤ a < 1 {\displaystyle 0\leq a<1} and b > 0 {\displaystyle b>0} and (ii) ‖ S u ‖ ≤ a ‖ T u ‖ + b ‖ u ‖ {\displaystyle \|Su\|\leq a\|Tu\|+b\|u\|} for every u ∈ dom ( T ) {\displaystyle u\in \operatorname {dom} (T)} , then T + S {\displaystyle T+S} is closed. Proof: Suppose ‖ u j − u ‖ 1 + ‖ ( T + S ) u j − f ‖ 2 → 0 {\displaystyle \|u_{j}-u\|_{1}+\|(T+S)u_{j}-f\|_{2}\to 0} . Then ‖ T ( u j − u k ) ‖ ≤ ‖ ( T + S ) ( u j − u k ) ‖ + a ‖ T ( u j − u k ) ‖ + b ‖ u j − u k ‖ {\displaystyle \|T(u_{j}-u_{k})\|\leq \|(T+S)(u_{j}-u_{k})\|+a\|T(u_{j}-u_{k})\|+b\|u_{j}-u_{k}\|} Thus, ( 1 − a ) ‖ T ( u j − u k ) ‖ ≤ ‖ ( T + S ) ( u j − u k ) ‖ + b ‖ u j − u k ‖ {\displaystyle (1-a)\|T(u_{j}-u_{k})\|\leq \|(T+S)(u_{j}-u_{k})\|+b\|u_{j}-u_{k}\|} By hypothesis, the right-hand side goes to 0 {\displaystyle 0} as j , k → ∞ {\displaystyle j,k\to \infty } . Since T {\displaystyle T} is closed, ( u j , T u j ) {\displaystyle (u_{j},Tu_{j})} converges to ( u , T u ) {\displaystyle (u,Tu)} . ◻ {\displaystyle \square } In particular, with a = 0 {\displaystyle a=0} , the hypothesis of the theorem is fulfilled, if S {\displaystyle S} is continuous. When X , Y {\displaystyle {\mathcal {X}},{\mathcal {Y}}} are normed spaces, by L ( X , Y ) {\displaystyle L({\mathcal {X}},{\mathcal {Y}})} we denote the space of all continuous linear operators from X {\displaystyle {\mathcal {X}}} to Y {\displaystyle {\mathcal {Y}}} . 2 Theorem If Y {\displaystyle {\mathcal {Y}}} is complete, then every Cauchy sequence T n {\displaystyle T_{n}} in L ( X , Y ) {\displaystyle L({\mathcal {X}},{\mathcal {Y}})} converges to a limit T {\displaystyle T} and ‖ T ‖ = lim n → ∞ ‖ T n ‖ {\displaystyle \|T\|=\lim _{n\to \infty }\|T_{n}\|} . Conversely, if L ( X , Y ) {\displaystyle L({\mathcal {X}},{\mathcal {Y}})} is complete, then so is Y. Proof: Let T n {\displaystyle T_{n}} be a Cauchy sequence in operator norm. For each x ∈ X {\displaystyle x\in {\mathcal {X}}} , since ‖ T n ( x ) − T m ( x ) ‖ ≤ ‖ T n − T m ‖ ‖ x ‖ {\displaystyle \|T_{n}(x)-T_{m}(x)\|\leq \|T_{n}-T_{m}\|\|x\|} and Y {\displaystyle {\mathcal {Y}}} is complete, there is a limit y {\displaystyle y} to which T n ( x ) {\displaystyle T_{n}(x)} converges. Define T ( x ) = y {\displaystyle T(x)=y} . T {\displaystyle T} is linear since the limit operations are linear. It is also continuous since ‖ T ( x ) ‖ ≤ sup n ‖ T n ‖ ‖ x ‖ {\displaystyle \|T(x)\|\leq \sup _{n}\|T_{n}\|\|x\|} . Finally, lim n → ∞ ‖ T n − T ‖ = sup ‖ x ‖ ≤ 1 ‖ lim n → ∞ T n ( x ) − T ( x ) ‖ {\displaystyle \lim _{n\to \infty }\|T_{n}-T\|=\sup _{\|x\|\leq 1}\|\lim _{n\to \infty }T_{n}(x)-T(x)\|} and | ‖ T n ‖ − ‖ T ‖ | ≤ ‖ T n − T ‖ → 0 {\displaystyle |\|T^{n}\|-\|T\||\leq \|T^{n}-T\|\to 0} as n → ∞ {\displaystyle n\to \infty } . (TODO: a proof for the converse.) ◻ {\displaystyle \square } 2 Theorem (uniform boundedness principle) Let F {\displaystyle {\mathcal {F}}} be a family of continuous functions f : X → Y {\displaystyle f:X\to Y} where Y {\displaystyle Y} is a normed linear space. Suppose that M ⊂ X {\displaystyle M\subset X} is non-meager and that: sup { ‖ f ( x ) ‖ : f ∈ F } < ∞ {\displaystyle \sup\{\|f(x)\|:f\in {\mathcal {F}}\}<\infty } for each x ∈ M {\displaystyle x\in M} It then follows: there is some G ⊂ X {\displaystyle G\subset X} open and such that (a) sup { ‖ f ( x ) ‖ : f ∈ F , x ∈ G } < ∞ {\displaystyle \sup\{\|f(x)\|:f\in {\mathcal {F}},x\in G\}<\infty } If we assume in addition that each member of F {\displaystyle {\mathcal {F}}} is a linear operator and X {\displaystyle X} is a normed linear space, then (b) sup { ‖ f ‖ : f ∈ F } < ∞ {\displaystyle \sup\{\|f\|:f\in {\mathcal {F}}\}<\infty } Proof: Let E j = ∩ f ∈ F { x ∈ X ; ‖ f ( x ) ‖ ≤ j } {\displaystyle E_{j}=\cap _{f\in {\mathcal {F}}}\{x\in X;\|f(x)\|\leq j\}} be a sequence. By hypothesis, M ⊂ ⋃ j = 1 ∞ E j {\displaystyle M\subset \bigcup _{j=1}^{\infty }E_{j}} and each E j {\displaystyle E_{j}} is closed since { x ∈ X ; ‖ f ( x ) ‖ > j } {\displaystyle \{x\in X;\|f(x)\|>j\}} is open by continuity. It then follows that some E N {\displaystyle E_{N}} has an interior point y {\displaystyle y} ; otherwise, M {\displaystyle M} fails to be non-meager. Hence, (a) holds. To show (b), making additional assumptions, we can find an open ball B = B ( 2 r , y ) ⊂ E N {\displaystyle B=B(2r,y)\subset E_{N}} . It then follows: for any f ∈ F {\displaystyle f\in {\mathcal {F}}} and any x ∈ X {\displaystyle x\in X} with ‖ x ‖ = 1 {\displaystyle \|x\|=1} , ‖ f ( x ) ‖ = r − 1 ‖ f ( r x + y ) − f ( y ) ‖ ≤ 2 r − 1 N {\displaystyle \|f(x)\|=r^{-1}\|f(rx+y)-f(y)\|\leq 2r^{-1}N} . ◻ {\displaystyle \square } A family Γ {\displaystyle \Gamma } of linear operators is said to be equicontinuous if given any neighborhood W {\displaystyle W} of 0 {\displaystyle 0} we can find a neighborhood V {\displaystyle V} of 0 {\displaystyle 0} such that: f ( V ) ⊂ W {\displaystyle f(V)\subset W} for every f ∈ Γ {\displaystyle f\in \Gamma } The conclusion of the theorem, therefore, means that the family satisfying the hypothesis of the theorem is equicontinuous. 2 Corollary Let X , Y , Z {\displaystyle {\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}}} be Banach spaces. Let T : X × Y → Z {\displaystyle T:{\mathcal {X}}\times {\mathcal {Y}}\to {\mathcal {Z}}} be a bilinear or sesquilinear operator. If T {\displaystyle T} is separately continuous (i.e., the function is continuous when all but one variables are fixed) and Y {\displaystyle {\mathcal {Y}}} is complete, then T {\displaystyle T} is continuous. Proof: For each y ∈ Y {\displaystyle y\in {\mathcal {Y}}} , sup { ‖ T ( x , y ) ‖ Z ; ‖ x ‖ X ≤ 1 } = ‖ T ( ⋅ , y ) ‖ {\displaystyle \sup\{\|T(x,y)\|_{\mathcal {Z}};\|x\|_{\mathcal {X}}\leq 1\}=\|T(\cdot ,y)\|} where the right-hand side is finite by continuity. Hence, the application of the principle of uniform boundedness to the family { T ( x , ⋅ ) ; ‖ x ‖ X ≤ 1 } {\displaystyle \{T(x,\cdot );\|x\|_{\mathcal {X}}\leq 1\}} shows the family is equicontinuous. That is, there is K > 0 {\displaystyle K>0} such that: ‖ T ( x , y ) ‖ Z ≤ K ‖ y ‖ Y {\displaystyle \|T(x,y)\|_{\mathcal {Z}}\leq K\|y\|_{\mathcal {Y}}} for every ‖ x ‖ X l e 1 {\displaystyle \|x\|_{\mathcal {X}}le1} and every y ∈ Y {\displaystyle y\in {\mathcal {Y}}} .The theorem now follows since X × Y {\displaystyle {\mathcal {X}}\times {\mathcal {Y}}} is a metric space. ◻ {\displaystyle \square } Since scalar multiplication is a continuous operation in normed spaces, the corollary says, in particular, that every linear operator on finite dimensional normed spaces is continuous. The next is one more example of the techniques discussed so far. 2. Theorem (Hahn-Banach) Let ( X , ‖ ⋅ ‖ ) {\displaystyle ({\mathcal {X}},\|\cdot \|)} be normed space and M ⊂ X {\displaystyle {\mathcal {M}}\subset {\mathcal {X}}} be a linear subspace. If z {\displaystyle z} is a linear functional continuous on M {\displaystyle {\mathcal {M}}} , then there exists a continuous linear functional w {\displaystyle w} on X {\displaystyle {\mathcal {X}}} such that z = w {\displaystyle z=w} on M {\displaystyle {\mathcal {M}}} and ‖ z ‖ = ‖ w ‖ {\displaystyle \|z\|=\|w\|} . Proof: Apply the Hahn-Banach stated in Chapter 1 with ‖ z ‖ ‖ ⋅ ‖ {\displaystyle \|z\|\|\cdot \|} as a sublinear functional dominating z {\displaystyle z} . Then: ‖ z ‖ = sup { ‖ w ( x ) ‖ ; x ∈ M , ‖ x ‖ ≤ 1 } ≤ sup { ‖ w ( x ) ‖ ; x ∈ X , ‖ x ‖ ≤ 1 } = ‖ w ‖ ≤ ‖ z ‖ {\displaystyle \|z\|=\sup\{\|w(x)\|;x\in {\mathcal {M}},\|x\|\leq 1\}\leq \sup\{\|w(x)\|;x\in {\mathcal {X}},\|x\|\leq 1\}=\|w\|\leq \|z\|} ;that is, ‖ z ‖ = ‖ w ‖ {\displaystyle \|z\|=\|w\|} . ◻ {\displaystyle \square } 2. Corollary Let M {\displaystyle {\mathcal {M}}} be a subspace of a normed linear space X {\displaystyle {\mathcal {X}}} . Then x {\displaystyle x} is in the closure of M {\displaystyle {\mathcal {M}}} if and only if z ( x ) {\displaystyle z(x)} = 0 for any z ∈ X ∗ {\displaystyle z\in {\mathcal {X}}^{*}} that vanishes on M {\displaystyle {\mathcal {M}}} . Proof: By continuity z ( M ¯ ) ⊂ z ( M ) ¯ {\displaystyle z({\overline {\mathcal {M}}})\subset {\overline {z({\mathcal {M}})}}} . Thus, if x ∈ M ¯ {\displaystyle x\in {\overline {\mathcal {M}}}} , then z ( x ) ∈ z ( M ) ¯ = { 0 } {\displaystyle z(x)\in {\overline {z({\mathcal {M}})}}=\{0\}} . Conversely, suppose x ∉ M ¯ {\displaystyle x\not \in {\overline {\mathcal {M}}}} . Then there is a δ > 0 {\displaystyle \delta >0} such that ‖ y − x ‖ ≥ δ {\displaystyle \|y-x\|\geq \delta } for every y ∈ M {\displaystyle y\in {\mathcal {M}}} . Define a linear functional z ( y + λ x ) = λ {\displaystyle z(y+\lambda x)=\lambda } for y ∈ M {\displaystyle y\in {\mathcal {M}}} and scalars λ {\displaystyle \lambda } . For any λ ≠ 0 {\displaystyle \lambda \neq 0} , since − λ − 1 y ∈ M {\displaystyle -\lambda ^{-1}y\in {\mathcal {M}}} , | z ( y + λ x ) | = | λ | δ − 1 δ ≤ δ − 1 | λ | | λ − 1 y + x ‖ = ‖ y + λ x ‖ {\displaystyle |z(y+\lambda x)|=|\lambda |\delta ^{-1}\delta \leq \delta ^{-1}|\lambda ||\lambda ^{-1}y+x\|=\|y+\lambda x\|} .Since the inequality holds for λ = 0 {\displaystyle \lambda =0} as well, z {\displaystyle z} is continuous. Hence, in view of the Hahn-Banach theorem, z ∈ X {\displaystyle z\in {\mathcal {X}}} while we still have z = 0 {\displaystyle z=0} on M {\displaystyle {\mathcal {M}}} and z ( x ) ≠ 0 {\displaystyle z(x)\neq 0} . ◻ {\displaystyle \square } Here is a classic application. 2 Theorem Let X , Y {\displaystyle {\mathcal {X}},{\mathcal {Y}}} be Banach spaces, T : X → Y {\displaystyle T:{\mathcal {X}}\to {\mathcal {Y}}} be a linear operator. If x n → 0 {\displaystyle x_{n}\to 0} implies that ( z ∘ T ) x n → 0 {\displaystyle (z\circ T)x_{n}\to 0} for every z ∈ X ∗ {\displaystyle z\in {\mathcal {X}}^{*}} , then T {\displaystyle T} is continuous. Proof: Suppose x n → 0 {\displaystyle x_{n}\to 0} and T x n → y {\displaystyle Tx_{n}\to y} . For every z ∈ X ∗ {\displaystyle z\in {\mathcal {X}}^{*}} , by hypothesis and the continuity of z {\displaystyle z} , 0 = lim n → ∞ z ( T x n ) = z ( y ) {\displaystyle 0=\lim _{n\to \infty }z(Tx_{n})=z(y)} .Now, by the preceding corollary y = 0 {\displaystyle y=0} and the continuity follows from the closed graph theorem. ◻ {\displaystyle \square } 2 Theorem Let X {\displaystyle {\mathcal {X}}} be a Banach space. (i) Given E ⊂ X {\displaystyle E\subset {\mathcal {X}}} , E {\displaystyle E} is bounded if and only if sup E | f | < ∞ {\displaystyle \sup _{E}|f|<\infty } for every f ∈ X ∗ {\displaystyle f\in {\mathcal {X}}^{*}} (ii) Given x ∈ X {\displaystyle x\in {\mathcal {X}}} , if f ( x ) = 0 {\displaystyle f(x)=0} for every f ∈ X ∗ {\displaystyle f\in {\mathcal {X}}^{*}} , then x = 0 {\displaystyle x=0} .Proof: (i) By continuity, sup { | f ( x ) | ; x ∈ E } ≤ ‖ f ‖ sup E ‖ ⋅ ‖ {\displaystyle \sup\{|f(x)|;x\in E\}\leq \|f\|\sup _{E}\|\cdot \|} .This proves the direct part. For the converse, define T x f = f ( x ) {\displaystyle T_{x}f=f(x)} for x ∈ E , f ∈ X ∗ {\displaystyle x\in E,f\in {\mathcal {X}}^{*}} . By hypothesis | T x f | ≤ sup E | f | {\displaystyle |T_{x}f|\leq \sup _{E}|f|} for every x ∈ E {\displaystyle x\in E} .Thus, by the principle of uniform boundedness, there is K > 0 {\displaystyle K>0} such that: | T x f | ≤ K ‖ f ‖ {\displaystyle |T_{x}f|\leq K\|f\|} for every x ∈ E , f ∈ X ∗ {\displaystyle x\in E,f\in {\mathcal {X}}^{*}} Hence, in view of Theorem 2.something, for x ∈ E {\displaystyle x\in E} , ‖ x ‖ = sup { | f ( x ) | ; f ∈ X ∗ , ‖ f ‖ ≤ 1 } ≤ K {\displaystyle \|x\|=\sup\{|f(x)|;f\in {\mathcal {X}}^{*},\|f\|\leq 1\}\leq K} .(ii) Suppose x ≠ 0 {\displaystyle x\neq 0} . Define f ( s ( x ) ) = s ‖ x ‖ {\displaystyle f(s(x))=s\|x\|} for scalars s {\displaystyle s} . Now, f {\displaystyle f} is continuous since its domain is finite-dimensional, and so by the Hahn-Banach theorem we could extend the domain of f {\displaystyle f} in such a way we have f ∈ X ∗ {\displaystyle f\in {\mathcal {X}}^{*}} . ◻ {\displaystyle \square } 2. Corollary Let ( X , ‖ ⋅ ‖ ) {\displaystyle ({\mathcal {X}},\|\cdot \|)} be Banach, f j ∈ X ∗ {\displaystyle f_{j}\in {\mathcal {X}}^{*}} and M ⊂ X {\displaystyle {\mathcal {M}}\subset {\mathcal {X}}} dense and linear. Then f j ( x ) → 0 {\displaystyle f_{j}(x)\to 0} for every x ∈ X {\displaystyle x\in {\mathcal {X}}} if and only if sup j ‖ f j ‖ < ∞ {\displaystyle \sup _{j}\|f_{j}\|<\infty } and f j ( y ) → 0 {\displaystyle f_{j}(y)\to 0} for every y ∈ M {\displaystyle y\in {\mathcal {M}}} . Proof: Since f j {\displaystyle f_{j}} is Cauchy, it is bounded. This shows the direct part. To show the converse, let x ∈ X {\displaystyle x\in {\mathcal {X}}} . If y j ∈ M {\displaystyle y_{j}\in {\mathcal {M}}} , then | f j ( x ) | ≤ | f j ( x − y j ) | + | f j ( y j ) | ≤ ( sup j ‖ f j ‖ ) ‖ x − y j ‖ + | f j ( y j ) | {\displaystyle |f_{j}(x)|\leq |f_{j}(x-y_{j})|+|f_{j}(y_{j})|\leq (\sup _{j}\|f_{j}\|)\|x-y_{j}\|+|f_{j}(y_{j})|} By denseness, we can take y j {\displaystyle y_{j}} so that ‖ y j − x ‖ → 0 {\displaystyle \|y_{j}-x\|\to 0} . ◻ {\displaystyle \square } 2 Theorem Let T {\displaystyle T} be a continuous linear operator into a Banach space. If ‖ I − T ‖ < 1 {\displaystyle \|I-T\|<1} where I {\displaystyle I} is the identity operator, then the inverse T − 1 {\displaystyle T^{-1}} exists, is continuous and can be written by: T − 1 ( x ) = ∑ k = 0 ∞ ( I − T ) k ( x ) {\displaystyle T^{-1}(x)=\sum _{k=0}^{\infty }\left(I-T\right)^{k}(x)} for each x {\displaystyle x} in the range of T {\displaystyle T} .Proof: For n ≥ m {\displaystyle n\geq m} , we have: ‖ ∑ k = m n ( I − T ) k ( x ) ‖ ≤ ‖ x ‖ ∑ k = m n ‖ I − T ‖ k {\displaystyle \|\sum _{k=m}^{n}\left(I-T\right)^{k}(x)\|\leq \|x\|\sum _{k=m}^{n}\left\|I-T\right\|^{k}} .Since the series is geometric by hypothesis, the right-hand side is finite. Let S n = ∑ k = 0 n ( I − T ) k {\displaystyle S_{n}=\sum _{k=0}^{n}\left(I-T\right)^{k}} . By the above, each time x {\displaystyle x} is fixed, S n ( x ) {\displaystyle S_{n}(x)} is a Cauchy sequence and the assumed completeness implies that the sequence converges to the limit, which we denote by S ( x ) {\displaystyle S(x)} . Since for each x {\displaystyle x} sup n ≥ 1 ‖ S n ( x ) ‖ < ∞ {\displaystyle \sup _{n\geq 1}\|S_{n}(x)\|<\infty } , it follows from the principle of uniform boundedness that: sup n ≥ 1 ‖ S n ‖ ≤ ∞ {\displaystyle \sup _{n\geq 1}\|S_{n}\|\leq \infty } .Thus, by the continuity of norms, ‖ S ( x ) ‖ = lim n → ∞ ‖ S n ( x ) ‖ ≤ ( sup n ≥ 1 ‖ S n ‖ ) ‖ x ‖ {\displaystyle \|S(x)\|=\lim _{n\to \infty }\|S_{n}(x)\|\leq (\sup _{n\geq 1}\|S_{n}\|)\|x\|} .This shows that S {\displaystyle S} is a continuous linear operator since the linearity is easily checked. Finally, ‖ T S ( x ) − x ‖ = ‖ lim n → ∞ − ( I − T ) n + 1 ( x ) ‖ ≤ ‖ x ‖ lim n → ∞ ‖ I − T ‖ n + 1 = 0 {\displaystyle \|TS(x)-x\|=\|\lim _{n\to \infty }-(I-T)^{n+1}(x)\|\leq \|x\|\lim _{n\to \infty }\|I-T\|^{n+1}=0} .Hence, S {\displaystyle S} is the inverse to T {\displaystyle T} . ◻ {\displaystyle \square } 2 Corollary The space of invertible continuous linear operators X {\displaystyle {\mathcal {X}}} is an open subspace of L ( X , X ) {\displaystyle L({\mathcal {X}},{\mathcal {X}})} . Proof: If T ∈ L ( X , X ) {\displaystyle T\in L({\mathcal {X}},{\mathcal {X}})} and ‖ S − T ‖ < 1 ‖ T − 1 ‖ {\displaystyle \|S-T\|<{1 \over \|T^{-1}\|}} , then S {\displaystyle S} is invertible. ◻ {\displaystyle \square } If F {\displaystyle \mathbf {F} } is a scalar field and X {\displaystyle {\mathcal {X}}} is a normed space, then L ( X , F ) {\displaystyle L({\mathcal {X}},\mathbf {F} )} is called a dual of X {\displaystyle {\mathcal {X}}} and is denoted by X ∗ {\displaystyle {\mathcal {X}}^{*}} . In view of Theorem 2.something, it is a Banach space. A linear operator T {\displaystyle T} is said to be a compact operator if the image of the open unit ball under T {\displaystyle T} is relatively compact. We recall that if a linear operator between normed spaces maps bounded sets to bounded sets, then it is continuous. Thus, every compact operator is continuous. 2 Theorem Let X {\displaystyle {\mathcal {X}}} be a reflexive Banach space and Y {\displaystyle {\mathcal {Y}}} be a Banach space. Then a linear operator T : X → Y {\displaystyle T:{\mathcal {X}}\to {\mathcal {Y}}} is a compact operator if and only if T {\displaystyle T} sends weakly convergent sequence to norm convergent ones. Proof: Let x n {\displaystyle x_{n}} converges weakly to 0 {\displaystyle 0} , and suppose T x n {\displaystyle Tx_{n}} is not convergent. That is, there is an ϵ > 0 {\displaystyle \epsilon >0} such that T x n ≥ ϵ {\displaystyle Tx_{n}\geq \epsilon } for infinitely many n {\displaystyle n} . Denote this subsequence by y n {\displaystyle y_{n}} . By hypothesis we can then show (TODO: do this indeed) that it contains a subsequence y n k {\displaystyle y_{n_{k}}} such that T y n k {\displaystyle Ty_{n_{k}}} converges in norm, which is a contradiction. To show the converse, let E {\displaystyle E} be a bounded set. Then since X {\displaystyle {\mathcal {X}}} is reflexive every countable subset of E {\displaystyle E} contains a sequence x n {\displaystyle x_{n}} that is Cauchy in the weak topology and so by the hypothesis T x n {\displaystyle Tx_{n}} is a Cauchy sequence in norm. Thus, T ( E ) {\displaystyle T(E)} is contained in a compact subset of Y {\displaystyle {\mathcal {Y}}} . ◻ {\displaystyle \square } 2 Corollary (i) Every finite-rank linear operator T {\displaystyle T} (i.e., a linear operator with finite-dimensional range) is a compact operator. (ii) Every linear operator T {\displaystyle T} with the finite-dimensional domain is continuous.Proof: (i) is clear, and (ii) follows from (i) since the range of a linear operator has dimension less than that of the domain. ◻ {\displaystyle \square } 2 Theorem The set of all compact operators into a Banach space forms a closed subspace of the set of all continuous linear operators in operator norm. Proof: Let T {\displaystyle T} be a linear operator and ω {\displaystyle \omega } be the open unit ball in the domain of T {\displaystyle T} . If T {\displaystyle T} is compact, then T ( ω ¯ ) {\displaystyle T({\overline {\omega }})} is bounded (try scalar multiplication); thus, T {\displaystyle T} is continuous. Since the sum of two compacts sets is again compact, the sum of two compact operators is again compact. For the similar reason, α T {\displaystyle \alpha T} is compact for any scalar α {\displaystyle \alpha } . We conclude that the set of all compact operators, which we denote by E {\displaystyle E} , forms a subspace of continuous linear operators. To show the closedness, suppose S {\displaystyle S} is in the closure of E {\displaystyle E} . Let ϵ > 0 {\displaystyle \epsilon >0} be given. Then there is some compact operator T {\displaystyle T} such that ‖ S − T ‖ < ϵ / 2 {\displaystyle \|S-T\|<\epsilon /2} . Also, since T {\displaystyle T} is a compact operator, we can cover T ( ω ) {\displaystyle T(\omega )} by a finite number of open balls of radius ϵ / 2 {\displaystyle \epsilon /2} centered at z 1 , z 2 , . . . z n {\displaystyle z_{1},z_{2},...z_{n}} , respectively. It then follows: for x ∈ ω {\displaystyle x\in \omega } , we can find some j {\displaystyle j} so that ‖ T x − z j ‖ < ϵ / 2 {\displaystyle \|Tx-z_{j}\|<\epsilon /2} and so ‖ S x − T x ‖ ≤ ‖ S x − z j ‖ + ‖ z j − T x ‖ < ϵ {\displaystyle \|Sx-Tx\|\leq \|Sx-z_{j}\|+\|z_{j}-Tx\|<\epsilon } . This is to say, S ( ω ) {\displaystyle S(\omega )} is totally bounded and since the completeness its closure is compact. ◻ {\displaystyle \square } 2 Corollary If T n {\displaystyle T_{n}} is a sequence of compact operators which converges in operator norm, then its limit is a compact operator. 2 Theorem (transpose) Let X , Y {\displaystyle {\mathcal {X}},{\mathcal {Y}}} be Banach spaces, and u : X → Y {\displaystyle u:{\mathcal {X}}\to {\mathcal {Y}}} be a continuous linear operator. Define t u : Y ∗ → X ∗ {\displaystyle {}^{t}\!u:{\mathcal {Y}}^{*}\to {\mathcal {X}}^{*}} by the identity t u ( z ) ( x ) = u ( z ( x ) ) {\displaystyle {}^{t}\!u(z)(x)=u(z(x))} . Then t u {\displaystyle {}^{t}\!u} is continuous both in operator norm and the weak-* topology, and ‖ t u ‖ = ‖ u ‖ {\displaystyle \|{}^{t}\!u\|=\|u\|} . Proof: For any z ∈ Y ∗ {\displaystyle z\in {\mathcal {Y}}^{*}} ‖ t u ( z ) ‖ = sup ‖ x ‖ ≤ 1 | ( u ∘ z ) ( x ) | ≤ ‖ u ‖ ‖ z ‖ {\displaystyle \|{}^{t}\!u(z)\|=\sup _{\|x\|\leq 1}|(u\circ z)(x)|\leq \|u\|\|z\|} Thus, ‖ t u ‖ ≤ ‖ u ‖ {\displaystyle \|{}^{t}\!u\|\leq \|u\|} and t u {\displaystyle {}^{t}\!u} is continuous in operator norm. To show the opposite inequality, let ϵ > 0 {\displaystyle \epsilon >0} be given. Then there is x 0 ∈ X {\displaystyle x_{0}\in {\mathcal {X}}} with ( 1 − ϵ ) ‖ u ‖ ≤ | u ( x 0 ) | {\displaystyle (1-\epsilon )\|u\|\leq |u(x_{0})|} . Using the Hahn-Banach theorem we can also find ‖ z 0 ‖ = 1 {\displaystyle \|z_{0}\|=1} and z 0 ( u ( x 0 ) ) = | u ( x 0 ) | {\displaystyle z_{0}(u(x_{0}))=|u(x_{0})|} . Hence, ‖ t u ‖ = sup ‖ z ‖ ≤ 1 ‖ t u ( z ) ‖ ≥ ‖ t u ( z 0 ) ‖ = | z 0 ( u ( x ) ) | = | u ( x 0 ) | ≥ ( 1 − ϵ ) ‖ u ‖ {\displaystyle \|{}^{t}\!u\|=\sup _{\|z\|\leq 1}\|{}^{t}\!u(z)\|\geq \|{}^{t}\!u(z_{0})\|=|z_{0}(u(x))|=|u(x_{0})|\geq (1-\epsilon )\|u\|} .We conclude ‖ t u ‖ = ‖ u ‖ {\displaystyle \|{}^{t}\!u\|=\|u\|} . To show weak-* continuity let V {\displaystyle V} be a neighborhood of 0 {\displaystyle 0} in X ∗ {\displaystyle {\mathcal {X}}^{*}} ; that is, V = { z ; z ∈ X ∗ , | z ( x 1 ) | < ϵ , . . . , | z ( x n ) | < ϵ } {\displaystyle V=\{z;z\in {\mathcal {X}}^{*},|z(x_{1})|<\epsilon ,...,|z(x_{n})|<\epsilon \}} for some ϵ > 0 , x 1 , . . . , x n ∈ X {\displaystyle \epsilon >0,x_{1},...,x_{n}\in {\mathcal {X}}} . If we let y j = u ( x j ) {\displaystyle y_{j}=u(x_{j})} , then t u ( { z ; z ∈ Y ∗ , | z ( y 1 ) | < ϵ , . . . , | z ( y n ) | < ϵ } ) ⊂ V {\displaystyle {}^{t}\!u(\{z;z\in {\mathcal {Y}}^{*},|z(y_{1})|<\epsilon ,...,|z(y_{n})|<\epsilon \})\subset V} since z ( y j ) = t u ( z ) ( x j ) {\displaystyle z(y_{j})={}^{t}\!u(z)(x_{j})} . This is to say, t u {\displaystyle {}^{t}\!u} is weak-* continuous. ◻ {\displaystyle \square } 2 Theorem Let T : X → Y {\displaystyle T:{\mathcal {X}}\to {\mathcal {Y}}} be a linear operator between normed spaces. Then T {\displaystyle T} is compact if and only if its transpose T ′ {\displaystyle T'} is compact. Proof: Let K {\displaystyle K} be the closure of the image of the closed unit ball under T {\displaystyle T} . If T is compact, then K is compact. Let y n ∈ Y ∗ {\displaystyle y_{n}\in Y^{*}} be a bounded sequence. Then the restrictions of y n {\displaystyle y_{n}} to K is a bounded equicontinuous sequence in C ( K ) {\displaystyle C(K)} ; thus, it has a convergent subsequence y n k {\displaystyle y_{n_{k}}} by Ascoli's theorem. Thus, T ′ y n k ( x ) = y n k ( T x ) {\displaystyle T'y_{n_{k}}(x)=y_{n_{k}}(Tx)} is convergent for every x with ‖ x ‖ X ≤ 1 {\displaystyle \|x\|_{\mathcal {X}}\leq 1} , and so T ′ y n k {\displaystyle T'y_{n_{k}}} is convergent. The converse follows from noting that every normed space can be embedded continuously into its second dual. (TODO: need more details.) ◻ {\displaystyle \square } == References ==