Preliminaries on normed vector space

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Preliminaries on normed vector space
Enormed vector space
topological dual of E i.e.
is the set of all continuous linear functionals
on E
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Continuous linear functional
normed vector space
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is a Banach space
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Propositions about normed vector space
1. If E is a normed vector space, then
is a Banach space
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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.
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I.2 Geometric form of Hahn-Banach Theorem

• separation of convex set

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Hyperplane
Ereal vector space
is called a Hyperplane of equationfa
If a0, then H is a Hypersubspace
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Proposition 1.5
E real normed vector space
The Hyperplane fa is closed
if and only if
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Separated in broad sense
Ereal vector space
A,B subsets of E
A and B are separated by the Hyperplanefa in
broad sense if
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Separated in restrict sense
Ereal vector space
A,B subsets of E
A and B are separated by the Hyperplanefa in
restrict sense if
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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.
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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.
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Corollary 1.8
Ereal normed vector space
Let F be a subspace of E with
,then

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

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