Preliminaries on normed vector space E normed vector

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

Continuous linear functional : normed vector space

is a Banach space

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

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.

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

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

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

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

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.

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.

Corollary 1. 8 E: real normed vector space Let F be a subspace of E with , then

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