1. Chaper 3 Weak Topologies. Reflexive Space.Separabe Space. Uniform Convex Spaces.

2. III.1

3. The weakest Topology Recall on the weakest topology which renders a family of mapping continuous topological space arbitary set

4. To define the weakest topology on X such that is continuous from X to for each Let must be open in X

5. For any finite set (*) : open in The family of the sets of the form (*) form a base of a topology F of X The topology is the weakest topology that renders allcontinuous

6. Proposition III.1 Letbe a sequence in X, then F F( )

9. Proposition III.2 Let Z be a topological space and Thenis continuous is continuous from Z to

11. III.2 Definition and properties of the weak topology σ(E,E´)

12. Definition σ(E,E´) E: Banach space E´: topological dual of E see next page

13. Definition : The weak topology is the weakest topology on E such that is continuous for each

14. Proposition III.3 The topology on E is Hausdorff

16. Proposition III.4 Let; we obtain a base of neighborhood of by consider sets of the form where, and F is finite

17. Proposition III.5 Letbe a sequence in E. Then (i) (ii) ifstrongly, then weakly.

18. (iii) if weakly, then is bounded and

19. (iv) if weakly and strongly in E´, then

22. Exercise Let E, F be real normed vector space consider on E and F the topologies and Then the product topology on E X F is respectively.

23. Proposition III.6 If,then is strong topology on E.

26. Remark If,then is strictly weaker then the strong topology.

29. III.3 Weak topology, convex set and linear operators

30. Theorem III.7 Letbe convex, then C is weakly closed if and only if C is strongly closed.

32. Remark The proof actually show that every every strongly closed convex set is an intersection of closed half spaces

33. Corollary III.8 If is convex l.s.c. w.r.t. strongly topology then In particular, if is l.s.c. w.r.t. then

36. Theorem III.9 Let E and F be Banach spaces and let be linear continuous (strongly), then T is linear continuous on E with to F with And conversely.

39. Remark Onis weak topology by

40. In genernal j is not surjective E is called reflexive If

41. III.4 The weak* topology σ(E′,E)

42. The weak* topology is the weakest topology on E´ such that is continuous for all

43. Proposition III.10 The weak* topology on E´ is Hausdorff

44. Proposition III.11 One obtains a base of a nhds for a by considering sets of the form

45. Proposition III.12 Let (i) be a sequence in E´, then

46. (ii) If strongly, then

47. (iii) If then

48. (iv) If then is bounded and

49. (v) If and strongly, then

50. Lemma III.2 Let X be a v.s. and are linear functionals´on X such that

52. Proposition III.13 If then there is is linear continuous´w.r.t

54. Corollary III.14 If H is a hyperplane in E´ closed w.r.t Then H is of the form

57. III.5 Reflexive spaces

58. Remark Onis weak topology by

59. j is isometry j(E) is closed vector subspace of

60. In genernal j is not surjective E is called reflexive If

61. Lemma 1 (Helly) p.1 Let E be a Banach space, are fixed. and Then following statements are equivalent

62. Lemma 1 (Helly) p.2 (i) (ii) where

66. Lemma 2 (Goldstine) Let E be a Banach space. Then is dense in w.r.t the weak* topology

68. Theorem (Banach Alaoglu-Bornbaki) is compact w.r.t.

69. Theorem A Banach space E is reflexive if and only if is compact w.r.t weak topology

72. Exercise Suppose that E is a reflexive Banach space. Show that evere closed vector subspace M of E is reflexive.

75. Corollary 1 Let E be a Banach space. Then E is reflexive if and only if is reflexive

77. Corollary 2 Let E be a reflexive Banach space. Suppose that if K is closed convex and bounded subset of E. Then K is compact w.r.t

79. Uniformly Convex A Banach space is called uniform convex if for all ε>0, there is δ>0 such that if

80. x y

81. Counter Example for Uniformly Convex Consider is not uniform convex. see next page

82. x y (0,1) (1,0) (0,-1) (-1,0)

83. Example for Uniformly Convex Consider is uniform convex. see next page

85. Theorem A uniformly convex Banach space E is reflexive.