Applications of Hahn Banach Theorem E normed vector

Applications of Hahn Banach Theorem

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

Corollary 1 Proof:

Corollary 2 Proof in next page This corollary implies that We may consider E as embedded in as normed space, then complete space which is the completion of E. is a

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 Let then be as in Mazur-Orlicz Theorem

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

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 Let E be a real reflexive Banach space. For (see next page) 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 topological linear space. Define+constant 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 with there is a unique such that

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.