Preliminaries on normed vector space

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Presentation transcript:

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