Theorem of Banach stainhaus and of Closed Graph Chapter 2 Theorem of Banach stainhaus and of Closed Graph

II.1Recall of Baire’s Lemma

Lemma II.1 (Baire) Let X be a complete metric space and a seq. of closed sets. Assume that for each n . Then

Baire’s Category Theorem Remark 1 Baire’s Category Theorem Baire’s Lemma is usually used in the following form. Let X be a nonempty complete metric space and a seq. of closed sets such that . Then there is such that

II.2 The Theorem of Banach -Steinhaus

L(E,F) L(E,E)=L(E) Let E and F be two normed vector spaces. Denoted by L(E,F) the space of all linear continuous operators from E to F equiped with norm L(E,E)=L(E)

Theorem II.1(Banach Steinhaus) Let E and F be two Banach space and a family of linear continuous operators from E to F Suppose (1) then (2) In other words, there is c such that

Remark 2 In American literature, Theorem II.1 is referred as principle of uniform boundness, which expresses well the conceit of the result: One deduces a uniform estimate from pointwise estimates.

Corollary II.2 Let E and F be two Banach spaces and a family of linear continuous operators from E to F such that for each converges as to a limit denoted by Tx. Then we have

Corollary II.3 Let G be a Banach space and B a subset of G. Suppose that (3) For all f , the set is bounded.(in R) Then (4) B is bounded

Dual statement of corollary II.3 Let G be a Banach space and a subset of . Suppose that (5) For all , the set is bounded. Then (6) is bounded.

II.3 Open Mapping Theorem And Closed Graph Theorem

Theorem II.5 (Open Mapping Thm,Banach) Let E and F be two Banach spaces and T a surjective linear continuous from E onto F. Then there is a constant c>0 such that

Remark 4 Property (7) implies that T maps each open set in E into open set in F (hence the name of the Theorem) In fact, let U be an open set in E, let us prove that TU is open in T.

Corollary II.6 Let E and F be Banach spaces be linear continuous and bijective. Then is continuous from F to E

Argument by homogeneity

Remark 5 Let E be a vector space equiped with two norms and Assume that and are Banach space and assume that there is C such that Then there is c>0 such that

i.e. and are equivalent Proof: Apply Corolary II.6 with and T is identity

Graph G(T) The graph G(T) of a linear operator from E to F is the set

Theorem II.7 (Closed Graph Theorem) Let E and F be two Banach spaces and T a linear operator from E to F. Suppose that the graph G(T) is closed in Then T is continuous. (converse also holds)