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Glossary of symbols for the table below:
- denotes the field of real numbers or complex numbers
- is a compact Hausdorff space.
- are real numbers with that are Hölder conjugates, meaning that they satisfy and thus also
- is a -algebra of sets.
- is an algebra of sets (for spaces only requiring finite additivity, such as the ba space).
- is a measure with variation A positive measure is a real-valued positive set function defined on a -algebra which is countably additive.
Classical Banach spaces | ||||||
Dual space | Reflexive | weakly sequentially complete | Norm | Notes | ||
---|---|---|---|---|---|---|
Yes | Yes | Euclidean space | ||||
Yes | Yes | |||||
Yes | Yes | |||||
Yes | Yes | |||||
No | Yes | |||||
No | No | |||||
No | No | |||||
No | No | Isomorphic but not isometric to | ||||
No | Yes | Isometrically isomorphic to | ||||
No | Yes | Isometrically isomorphic to | ||||
No | No | Isometrically isomorphic to | ||||
No | No | Isometrically isomorphic to | ||||
No | No | |||||
No | No | |||||
? | No | Yes | ||||
? | No | Yes | A closed subspace of | |||
? | No | Yes | A closed subspace of | |||
Yes | Yes | |||||
No | Yes | The dual is if is -finite. | ||||
? | No | Yes | is the total variation of | |||
? | No | Yes | consists of functions such that | |||
No | Yes | Isomorphic to the Sobolev space | ||||
No | No | Isomorphic to essentially by Taylor's theorem. |
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