List of Banach spaces

In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to be Banach spaces as well.

Classical Banach spaces

According to harvtxt|Diestel|1984|loc=Chapter VII, the classical Banach spaces are those defined by harvtxt|Dunford|Schwartz|1958, which is the source for the following table.

Here K denotes the field of real numbers or complex numbers and "I" is a closed and bounded interval ["a","b"] . The number "p" is a real number with 1<"p"<∞, and "q" is its Hölder conjugate:

:$q=frac\{p\}\{p-1\}.$

The symbol &Sigma; denotes a &sigma;-algebra of sets, and &Xi; denotes just an algebra of sets (for spaces only requiring finite additivity, such as the ba space). The symbol &mu; denotes a positive measure: that is, a real-valued positive set function defined on a &sigma;-algebra which is countably additive.

Banach spaces in other areas of analysis

* The Hardy spaces
* The space "BMO" of functions of bounded mean oscillation
* The space of functions of bounded variation
* Sobolev spaces
* The Birnbaum–Orlicz spaces "L"_A(&mu;).
* Hölder spaces C^k,α(&Omega;).

Notes

References

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