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 Diestel (1984, Chapter VII), the classical Banach spaces are those defined by Dunford & Schwartz (1958), which is the source for the following table. Here K denotes the field of real numbers or complex numbers, p is a real number with 1<p<∞, and q is its Hölder conjugate:

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

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

Classical Banach spaces
	Dual space	Reflexive	weakly complete	Norm	Notes
Kn	Kn	Yes	Yes	$\|x\|_{2}=\left(\sum _{i=1}^{n}|x_{i}|^{2}\right)^{1/2}$
ℓnp	ℓnq	Yes	Yes	$\|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{1/p}$
ℓn∞	ℓn1	Yes	Yes	$\|x\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|$
ℓp	ℓq	Yes	Yes	$\|x\|_{p}=\left(\sum _{i=1}^{\infty }|x_{i}|^{p}\right)^{1/p}$
ℓ1	ℓ∞	No	Yes	$\|x\|_{1}=\sum _{i=1}^{\infty }|x_{i}|$
ℓ∞	ba	No	No	$\|x\|_{\infty }=\sup _{i}|x_{i}|$
c	ℓ1	No	No	$\|x\|_{\infty }=\sup _{i}|x_{i}|$
c0	ℓ1	No	No	$\|x\|_{\infty }=\sup _{i}|x_{i}|$
bv	ℓ1 + K	No	Yes	$\|x\|_{bv}=|x_{1}|+\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|$
bv0	ℓ1	No	Yes	$\|x\|_{bv_{0}}=\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|$
bs	ba	No	No	$\|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|$	Isometrically isomorphic to ℓ∞.
cs	ℓ1	No	No	$\|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|$	Isometrically isomorphic to c.
B(X, Ξ)	ba(Ξ)	No	No	$\|f\|_{B}=\sup _{x\in X}|f(x)|$
C(X)	rca(X)	No	No	$\|x\|_{bs}=\sup _{x\in X}\left|f(x)\right|$	X is a compact Hausdorff space.
ba(Ξ)	?	No	Yes	$\|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)$ (variation of a measure)
ca(Σ)	?	No	Yes	$\|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)$
rca(Σ)	?	No	Yes	$\|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)$
Lp(μ)	Lq(μ)	Yes	Yes	$\|f\|_{p}=\left\{\int |f|^{p}\,d\mu \right\}^{1/p}$

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

References

• Diestel, Joseph (1984), Sequences and series in Banach spaces, Springer-Verlag, ISBN 0-387-90859-5.
• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.