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
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- 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