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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#### Theorem 1.4 Hahn-Banach 2nd Geometric Form

#### Result

This makes it possible for us to use a closed hyperplane to strictly separate a closed convex subset and a compact convex subset where no common areas of them could be found.

#### Remark

#### Proof(See Appendix Above)

The proof of Theorem 1.4 is by setting C = A – B and three claims (1)C is closed (2)C is convex (3)0 does not live inside C, then apply the properties of open balls and the result of Theorem 1.3

#### Reference

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

#### Related Topics

- Definitions
- Convex
- Closed
- Hyperplane
- Separate
- Compactness

- Theorems/Corollary/Propositions/Lemmas

### Corollary 1.4.1

#### Statement

#### Result

This gives us a possible choice in proving that F is dense in E, since it suffices to prove that everywhere continuous linear functional on E that vanishes on F must vanish everywhere on E.

#### Remark

#### Reference

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

#### Related Topics

- Definitions
- Closure
- Dense
- Hyperplane

- Theorems/Corollary/Propositions/Lemmas