FASSPDE-Chapter1-Propositions

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Proposition 1.2

Proof(See Appendix Above)

Hahn-Banach Theorem admits an extension of the linear functional on a linear Subspace to the whole space which is bounded by a maximal element admitted by the Zorn’s Lemma.

Reference

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

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Proposition 1.5

Proof(See Appendix Above)

This proposition provides a possible method to translate the closure of M in to the Annihilator of M. However, the double perp of the dual subspace may be strictly bigger than the closure of the dual itself.

Remark

Reference

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

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Proposition 1.6

Proof(See Appendix Above)

We therefore obtain a method which by saying convex l.s.c. phi with such phi identically not equal to infinity, we can always have its phi* identically not equal to infinity and bounded below.

Reference

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

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