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 topological space arbitary set

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

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

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

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

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

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

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

Proposition III.3 The topology on E is Hausdorff

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

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

(iii) if weakly, then is bounded and

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

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.

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

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

III.3 Weak topology, convex set and linear operators

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

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

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

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.

Remark Onis weak topology by

In genernal j is not surjective E is called reflexive If

III.4 The weak* topology σ(E′,E)

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

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

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

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

(ii) If strongly, then

(iii) If then

(iv) If then is bounded and

(v) If and strongly, then

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

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

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

III.5 Reflexive spaces

Remark Onis weak topology by

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

In genernal j is not surjective E is called reflexive If

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

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

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

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

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

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

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

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

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

x y

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

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

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

Theorem A uniformly convex Banach space E is reflexive.