Lemma II.1 (Baire)

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

2. 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

3. 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.

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

5. Theorem II.1(BanachSteinhaus) 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

6. Application of Banach Steinhaus

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

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

13. Lebesque Theorem such that

16. Dirichlet kernel

19. II.4 Topological Complementoperators invertible on right(resp. on left)

20. 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

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

24. 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

25. (14) L G x

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

30. 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 • L is closed. • G∩L={0} and G+L=E see next page

31. 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.

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

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

38. Remark 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

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

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

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

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

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