Chapter 2 Theorem of Banach stainhaus and of

Chapter 2 Theorem of Banach stainhaus and of Closed Graph

II. 1 Recall of Baire’s Lemma

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

Baire’s Category Remark 1 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 is such that . Then there

II. 2 Theorem of Banach Steinhaus

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

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

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 by Tx. Then we have to a limit denoted

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 Corollary II. 4 Let G be a Banach space and a subset of (5) For all . Suppose that , 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 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)