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Chaper 3

Chaper 3. Weak Topologies. Reflexive Space .Separabe Space. Uniform Convex Spaces. III.1. The weakest Topology. Recall on the weakest topology which renders a family of mapping continuous. arbitary set. topological space. To define the weakest topology on X such that.

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Chaper 3

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  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 arbitary set topological space

  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 Fof X The topology is the weakest topology that renders all continuous

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

  7. Proposition III.2 Let Z be a topological space and Then is continuous is continuous from Z to

  8. III.2 Definition and properties of the weak topologyσ(E,E´)

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

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

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

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

  13. Proposition III.5 Let be a sequence in E. Then (i) (ii) if strongly, then weakly.

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

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

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

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

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

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

  20. Theorem III.7 Let be convex, then C is weakly closed if and only if C is strongly closed.

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

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

  23. 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.

  24. Remark On is weak topology by

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

  26. III.4 The weak* topology σ(E′,E)

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

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

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

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

  31. (ii) If strongly, then

  32. (iii) If then

  33. (iv) If then is bounded and

  34. (v) If and strongly, then

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

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