In order for convenience in checking definition along learning in Math, I found it necessary to gather definitions of the same kind so that distinguishing the differences among them will no longer be a tedious job. This work is very hard to do even many materials are available, if you want to use for commercial use, please contact me for permission. Furthermore, the work is still long from finished, if you want to contribute for more or better definitions, please contact me.

Tianyu Zhang

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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

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

#### 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

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

#### 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

#### Related Topics

- Definitions
- Conjugate(of functions)
- Convex
- Continuous(Lower Semi Continuous)
- Bounded
- Affine(of functions)

- Theorems/Corollary/Propositions/Lemmas