Hahn-Banach, 1st Geometric Form(FASSPDE)

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Theorem 1.3 Hahn-Banach 1st Geometric Form

Result

This makes it possible for us to use a hyperplane to separate a convex set and a convex open set if they have empty intersection.

Proof(See Appendix Above)

The proof of Theorem 1.2 relies on the Lemma 1.2 and the Lemma 1.3

Reference

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Chapter 1

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Lemma 1.2

Statement

Result

This lemma makes it possible for us to construct a gauge function which (1) is able to be applied with the Hahn-Banach Theorem(2)bounded(3)generates the space

Proof(See Appendix Above)

(1)positive homogeneous is trivial. (3), (4)holds by the properties of open balls. (2) holds by the choice of t and arbitrary epsilon.

Reference

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Chapter 1

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Lemma 1.3

Statement

Result

Exists a closed hyperplane which separates a singleton and an open convex set.

Proof(See Appendix Above)

By the Hahn-Banach Theorem, we have the maximal p(x) and the extension f(x), and for some chosen value of x0 we have the separation.

Reference

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Chapter 1

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