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 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

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(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

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 where proved in next page

13. Lebesque Theorem such that

16. Dirichlet kernel

19. II.4 Topological Complement operators 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 GLx y z

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) GL x

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

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 (i)L is closed. (ii)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 for Topological 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. 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 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

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

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

50. 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

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

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