In order for convenience in checking definition along Learning in Math, I found it necessary to gather definitions of the same kind to distinguish the differences among them. This work is very hard to do even many materials are available; so if you want to use for commercial use, please contact me for permission, otherwise. 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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### 1. Bounded

#### Definition

#### Remark

One can check that a linear functional is continuous if and only if it is bounded.

#### Reference

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

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

## 2. Pointwise Bounded

#### Definition

#### Reference

Convex Analysis Rockafellar RT, Chapter2, Section10

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

## 3. Uniformly Bounded

#### Definition

#### Reference

Convex Analysis Rockafellar RT, Chapter2, Section10

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

## 4. Essentially Bounded

#### Definition

#### Reference

Measure Theory, Paul R. Halmos, Chapter 4 Section 21

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

## 5. Totally Bounded(of topological spaces)

#### Definition

#### Reference

https://ncatlab.org/nlab/show/totally+bounded+space

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas