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.
Variational Inequality (Stampachia-Hartmam) Then there is such that p. 2
Applications of Mazm. Orlich Theorem Inequality after mixing of functions
Theorem Let S be an arbitary set. The following two statements are equivalent:
Minimax Theorem of Von Neumann
Lemma VI. 1 (Riesz-Lemma) Let For any fixed , apply Green’s second identity to u and then let in the domain we have
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