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

Remark 1 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 Baire’s Category Theorem

First Category X: metric space, M is nonwhere dence in X i.e. has no ball in X. is nonwhere dense in X. M is called of first category.

By Baire’s Category Theorem No complement metric space is of first Category. is nonwhere dence in X.

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

Application of Banach Steinhaus

Fourier Series is called Fourier series of f is called Fourier nth partial sum of f

If f is real valued, then where proved in next page

Lebesque Theorem such that

Dirichlet kernel

II.4 Topological Complement operators invertible on right (resp. on left)

Theorem II.8 Let E be a Banach space and let G and L be two closed vector subspaces such that G+L is closed. Then there exists constant such that

(13) any element z of G+L admits a decomposition of the form z=x+y with GLx y z

Corollary II.9 Let E be a Banach space and let G and L be two closed vector subspaces such that G+L is closed. Then there exists constant such that

(14) GL x

Remark Let E be a Banach space and let G and L be two closed vector subspaces with Then G+L is closed. Exercise

Topological Complement Let G be a closed vector subspace of a Banach space E. A vector subspace L of E is called a topological complement of G if (i)L is closed. (ii)G∩L={0} and G+L=E see next page

In this case, all can be expressed uniquely as z=x+y with It follows from Thm II.8 that the projections z→x and z→y are linear continuous and surjective.

Example for Topological Complement E: Banach space G:finite dimensional subspace of E; hence is closed. Find a topological complement of G see next page

Remark On finite dimensional vector space, linear functional is continuous. Prove in next page

Let E be a Banach space. Let G be a closed v.s.s of E with codimG < ∞, then any algebraic complement is topological complement of G Typial example in next page

Let then be a closed vector subspace of E and codimG=p Prove in next page 證明很重要

Question Does there exist linear continuous map from F to E such that Let E and F be two Banach spaces is linear continuous surjective S is called an inverse on right of T

Theorem II.10 The folloowing properties are equivalent : Let E and F be two Banach spaces is linear continuous surjective

(i) T admits an inverse on right (ii) admits a topological complement Prove in next page

inverse on left If S is a linear continuous operator from F onto E such that Let E and F be two Banach spaces is linear continuous injective S is called an inverse on left of T

Theorem II.11 The following properties are equivalent : Let E and F be two Banach spaces is linear continuous injective

(i) T admits an inverse on left (ii) is closed and admits a topological complement. Prove in next page