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 1.3 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

#### Related Topics

- Definitions
- Convex
- Open
- Hyperplane
- Separate

- Theorems/Corollary/Propositions/Lemmas

### 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

#### Related Topics

- Definitions
- Convex
- Open
- Positive Homogeneous
- Additive(Subadditive)
- Gauge/Minkowski Functional

- Theorems/Corollary/Propositions/Lemmas

### 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

#### Related Topics

- Definitions
- Convex
- Open
- Separate
- Hyperplane

- Theorems/Corollary/Propositions/Lemmas