Applications of Hahn Banach Theorem

Applications of Hahn Banach Theorem

E: normed vector space, assumed to be real for definitions Known: Taking We have

Corollary1 Proof:

Corollary2 Proof in next page This corollary implies that We may consider E as embedded in as normed space, then is a complete space which is the completion of E.

A dual variational principle

Example Why? See next page

1 1 -1

Claim In this example

Exercise

Suppose that such that , then

Applications of Mazm- Orlich Theorem

is the space of the probability measure of S

Mazm-Orlich Theorem

Mazur-Orlicz (1953) S : arbitary set E : real vector space sublinear (I)

Corollary 1 be as in Mazur-Orlicz Theorem Let then

Corollary 2 If in Corollary 1 sastisfies the condition: For each there is such that then

Example p.1 S: arbitary set defined by Then

Example p.2 is q-convex

Example p.3 In particular, S is a convex set in a linear map is convex i.e. Then This implies von Neumann Minimax Theorem

Duality map p.1 J(x) is w-compact (see next page) Let E be a real reflexive Banach space. For J is a Duality map. If E is a Hilbert space, then

Lemma p.1 Let S be a compact convex subset of a linear function +constant topological linear space. Define If and F is the space of all affine functions then

Theorem p.1 Let E be a real reflexive Banach space is bilinear such that (i) There is c>0 such that (ii)

Theorem p.2 Then for each there is a unique such that with

Variational Inequality(Stampachia-Hartmam) p.1 E: reflexive Banach space C: closed bounded convex set in E satisfies (i) f is monotone i.e. (ii) f is weakly continuous on each line segment in C.

Variational Inequality(Stampachia-Hartmam) p.2 Then there is such that

Applications of Mazm- Orlich Theorem Inequality after mixing of functions

Theorem Let S be an arbitary set. The following two statements are equivalent:

Applications of Hahn Banach Theorem