Preliminary

Theorem I.1 Hahn-Banach, analytic form

Minkowski gauge Theorem Let K be a convex set in E with 0 being its interior point. Define a function

Proof of Minkowski gauge Theorem p.1

Proof of Minkowski gauge Theorem p.2

Minkowski gauge function of K is called the Minkowski gauge function of K

Lemma 1

Proof of Lemma 1 p.1

Proof of Lemma 1 p.2

Remark

Proof of Remark p.1

Hyperplane E:real vector space is called a Hyperplane of equation[f=α] If α=0, then H is a Hypersubspace

Proposition 1.5 E: real normed vector space The Hyperplane [f=α] is closed if and only if

Theorem 1.6(Hahn-Banach; the first geometric form) E:real normed vector space Let be two disjoint nonnempty convex sets. Suppose A is open, then there is a closed Hyperplane separating A and B in broad sense.

Epigraph E : set Epigraph, is the set

Conjugated function Assume that E is a real n.v.s Given such that Define the conjugated function of by

Theorem I.11 see next page Suppose and are convex and suppose that there is such that and is continuous at

Observe (1) usually appears for constrain (2) see next page

The proof of Thm I.11 see next page

Application of Thm I.11 Let be nonempty, close and convex. Put

Let

Application of Hahn-Banach Theorem E: real normed vector space G: vector subspace Then for any

Theorem II.5 (Open Mapping Thm,Banach) Let E and F be two Banach spaces and T a surjective linear continuous from E onto F. Then there is a constant c>0 such that

Theorem II.8 Let E be a Banach space and let G and L be two closed vector subspaces such that G+L is closed . Then there exists constant such that

(13) any element z of G+L admits a decomposition of the form z=x+y with L x G z y

Corollary II.9 Let E be a Banach space and let G and L be two closed vector subspaces such that G+L is closed . Then there exists constant such that

(14) L G x

II.5 Orthogonolity Relation

are closed vector subspace X: Banach space : vector subspace orthogonal of M : vector subspace Let are closed vector subspace of X and X’ ,respectively.

Proposition II.12 Suppose M is a vector subspace of X then If N is a vector subspace of then

Proposition II.13 Suppose G and L are closed vector subspaces of X. We have (16) (17)

Corollary II.14 Suppose G and L are closed vector subspaces of X. We have (18) (19)

Proposition II.15 p.1 Suppose G and L are closed vector subspaces of X. The following properties are equivalent:

Proposition II.15 p.2 (a) G+L is closed in X. (b) is closed in X´ (c)

II.5 Orthogonolity Relation

Theorem I.11 see next page Suppose and are convex and suppose that there is such that and is continuous at