Loading...

PPT – Preliminaries on normed vector space PowerPoint presentation | free to download - id: 693f6c-OTI2N

The Adobe Flash plugin is needed to view this content

Preliminaries on normed vector space

Enormed 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

(No Transcript)

(No Transcript)

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.

(No Transcript)

(No Transcript)

(No Transcript)

I.2 Geometric form of Hahn-Banach Theorem

- separation of convex set

Hyperplane

Ereal vector space

is called a Hyperplane of equationfa

If a0, then H is a Hypersubspace

Proposition 1.5

E real normed vector space

The Hyperplane fa is closed

if and only if

(No Transcript)

(No Transcript)

Separated in broad sense

Ereal vector space

A,B subsets of E

A and B are separated by the Hyperplanefa in

broad sense if

Separated in restrict sense

Ereal vector space

A,B subsets of E

A and B are separated by the Hyperplanefa in

restrict sense if

Theorem 1.6(Hahn-Banach the first geometric

form)

Ereal 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.

(No Transcript)

Theorem 1.7(Hahn-Banach the second geometric

form)

Ereal 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.

(No Transcript)

(No Transcript)

(No Transcript)

Corollary 1.8

Ereal normed vector space

Let F be a subspace of E with

,then

(No Transcript)

Exercise

A vector subspace F of E is dence

if and only if

(No Transcript)