Preliminary

1. Preliminary

2. Theorem I.1Hahn-Banach, analytic form

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

4. Proof of Minkowski gauge Theorem p.1

6. Proof of Minkowski gauge Theorem p.2

7. Minkowski gauge function of K • is called the Minkowski gauge function of K

8. Lemma 1

10. Proof of Lemma 1 p.1

11. Proof of Lemma 1 p.2

12. Remark

13. Proof of Remark p.1

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

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

18. 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.

20. Epigraph E : set Epigraph, is the set

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

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

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

26. The proof of Thm I.11 see next page

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

32. Let

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

36. 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

41. 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

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

45. 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

46. (14) L G x

50. II.5 Orthogonolity Relation