1. Preliminaries on normed vector space
E:normed vector space
:topological dual of E i.e.
is the set of all continuous linear functionals on E

2. Continuous linear functional
:normed vector space

3. is a Banach space

6. Propositions about normed vector space
1. If E is a normed vector space, then
is a Banach space

7. Propositions about normed vector space
2. If E is a finite dimensiional
normed vector space, then
E is or with Euclidean norm
topologically depending on whether
E is real or complex.

11. I.2 Geometric form of Hahn-Banach Theorem
separation of convex set

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

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

16. Separated in broad sense
E:real vector space
A,B: subsets of E
A and B are separated by the
Hyperplane[f=α] in broad sense if

17. Separated in restrict sense
E:real vector space
A,B: subsets of E
A and B are separated by the
Hyperplane[f=α] in restrict sense 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. Theorem 1.7(Hahn-Banach; the second geometric form)
E:real normed vector space
Let be two disjoint
nonnempty closed convex sets.
Suppose that B is compact,
then there is a closed Hyperplane
separating A and B in restric sense.

24. Corollary 1.8 E:real normed vector space
Let F be a subspace of E with ,then

26. Exercise
A vector subspace F of E is dence
if and only if