1. Preliminary

2. Theorem I.1 Hahn-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 of by

23. Theorem I.11 see next page Suppose and 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

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

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

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

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

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

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

68. II.5 Orthogonolity Relation

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