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

#### Theorem 1.8 Frenchel-Rockafellar Theorem

#### Result

If the effective domains of two convex functions have non-empty intersection with one of the functions is continuous at the intersection, then both the infimum of their sum, supremum of the minus of their sum, the max of their minus sum, and the minus of their mean are attainable.

#### Reference

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

#### Related Topics

- Definitions
- Convex
- Domain
- Continuous

- Theorems/Corollary/Propositions/Lemmas

### Lemma 1.4

#### Statement

#### Result

The interior of a convex subset is also convex. If such interior is non empty, then the closure of such a convex set coincides the closure of its interior.

#### Reference

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

#### Related Topics

- Definitions
- Convex
- Interior

- Theorems/Corollary/Propositions/Lemmas