List of 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.

Glossary of symbols for the table below:

• ${\displaystyle \mathbb {F} }$ denotes the field of real numbers ${\displaystyle \mathbb {R} }$ or complex numbers ${\displaystyle \mathbb {C} .}$
• ${\displaystyle K}$ is a compact Hausdorff space.
• ${\displaystyle p,q\in \mathbb {R} }$ are real numbers with ${\displaystyle 1<p,q<\infty }$ that are Hölder conjugates, meaning that they satisfy ${\displaystyle {\frac {1}{q}}+{\frac {1}{p}}=1}$ and thus also ${\displaystyle q={\frac {p}{p-1}}.}$
• ${\displaystyle \Sigma }$ is a ${\displaystyle \sigma }$-algebra of sets.
• ${\displaystyle \Xi }$ is an algebra of sets (for spaces only requiring finite additivity, such as the ba space).
• ${\displaystyle \mu }$ is a measure with variation ${\displaystyle |\mu |.}$ A positive measure is a real-valued positive set function defined on a ${\displaystyle \sigma }$-algebra which is countably additive.

Classical Banach spaces
	Dual space	Reflexive	weakly sequentially complete	Norm		Notes
${\displaystyle \mathbb {F} ^{n}}$	${\displaystyle \mathbb {F} ^{n}}$	Yes	Yes	${\displaystyle \|x\|_{2}}$	${\displaystyle =\left(\sum _{i=1}^{n}|x_{i}|^{2}\right)^{1/2}}$	Euclidean space
${\displaystyle \ell _{p}^{n}}$	${\displaystyle \ell _{q}^{n}}$	Yes	Yes	${\displaystyle \|x\|_{p}}$	${\displaystyle =\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}}$
${\displaystyle \ell _{\infty }^{n}}$	${\displaystyle \ell _{1}^{n}}$	Yes	Yes	${\displaystyle \|x\|_{\infty }}$	${\displaystyle =\max \nolimits _{1\leq i\leq n}|x_{i}|}$
${\displaystyle \ell ^{p}}$	${\displaystyle \ell ^{q}}$	Yes	Yes	${\displaystyle \|x\|_{p}}$	${\displaystyle =\left(\sum _{i=1}^{\infty }|x_{i}|^{p}\right)^{\frac {1}{p}}}$
${\displaystyle \ell ^{1}}$	${\displaystyle \ell ^{\infty }}$	No	Yes	${\displaystyle \|x\|_{1}}$	${\displaystyle =\sum _{i=1}^{\infty }\left|x_{i}\right|}$
${\displaystyle \ell ^{\infty }}$	${\displaystyle \operatorname {ba} }$	No	No	${\displaystyle \|x\|_{\infty }}$	${\displaystyle =\sup \nolimits _{i}\left|x_{i}\right|}$
${\displaystyle \operatorname {c} }$	${\displaystyle \ell ^{1}}$	No	No	${\displaystyle \|x\|_{\infty }}$	${\displaystyle =\sup \nolimits _{i}\left|x_{i}\right|}$
${\displaystyle c_{0}}$	${\displaystyle \ell ^{1}}$	No	No	${\displaystyle \|x\|_{\infty }}$	${\displaystyle =\sup \nolimits _{i}\left|x_{i}\right|}$	Isomorphic but not isometric to ${\displaystyle c.}$
${\displaystyle \operatorname {bv} }$	${\displaystyle \ell ^{\infty }}$	No	Yes	${\displaystyle \|x\|_{bv}}$	${\displaystyle =\left|x_{1}\right|+\sum _{i=1}^{\infty }\left|x_{i+1}-x_{i}\right|}$	Isometrically isomorphic to ${\displaystyle \ell ^{1}.}$
${\displaystyle \operatorname {bv} _{0}}$	${\displaystyle \ell ^{\infty }}$	No	Yes	${\displaystyle \|x\|_{bv_{0}}}$	${\displaystyle =\sum _{i=1}^{\infty }\left|x_{i+1}-x_{i}\right|}$	Isometrically isomorphic to ${\displaystyle \ell ^{1}.}$
${\displaystyle \operatorname {bs} }$	${\displaystyle \operatorname {ba} }$	No	No	${\displaystyle \|x\|_{bs}}$	${\displaystyle =\sup \nolimits _{n}\left|\sum _{i=1}^{n}x_{i}\right|}$	Isometrically isomorphic to ${\displaystyle \ell ^{\infty }.}$
${\displaystyle \operatorname {cs} }$	${\displaystyle \ell ^{1}}$	No	No	${\displaystyle \|x\|_{bs}}$	${\displaystyle =\sup \nolimits _{n}\left|\sum _{i=1}^{n}x_{i}\right|}$	Isometrically isomorphic to ${\displaystyle c.}$
${\displaystyle B(K,\Xi )}$	${\displaystyle \operatorname {ba} (\Xi )}$	No	No	${\displaystyle \|f\|_{B}}$	${\displaystyle =\sup \nolimits _{k\in K}|f(k)|}$
${\displaystyle C(K)}$	${\displaystyle \operatorname {rca} (K)}$	No	No	${\displaystyle \|x\|_{C(K)}}$	${\displaystyle =\max \nolimits _{k\in K}|f(k)|}$
${\displaystyle \operatorname {ba} (\Xi )}$	?	No	Yes	${\displaystyle \|\mu \|_{ba}}$	${\displaystyle =\sup \nolimits _{S\in \Sigma }|\mu |(S)}$
${\displaystyle \operatorname {ca} (\Sigma )}$	?	No	Yes	${\displaystyle \|\mu \|_{ba}}$	${\displaystyle =\sup \nolimits _{S\in \Sigma }|\mu |(S)}$	A closed subspace of ${\displaystyle \operatorname {ba} (\Sigma ).}$
${\displaystyle \operatorname {rca} (\Sigma )}$	?	No	Yes	${\displaystyle \|\mu \|_{ba}}$	${\displaystyle =\sup \nolimits _{S\in \Sigma }|\mu |(S)}$	A closed subspace of ${\displaystyle \operatorname {ca} (\Sigma ).}$
${\displaystyle L^{p}(\mu )}$	${\displaystyle L^{q}(\mu )}$	Yes	Yes	${\displaystyle \|f\|_{p}}$	${\displaystyle =\left(\int |f|^{p}\,d\mu \right)^{\frac {1}{p}}}$
${\displaystyle L^{1}(\mu )}$	${\displaystyle L^{\infty }(\mu )}$	No	Yes	${\displaystyle \|f\|_{1}}$	${\displaystyle =\int |f|\,d\mu }$	The dual is ${\displaystyle L^{\infty }(\mu )}$ if ${\displaystyle \mu }$ is ${\displaystyle \sigma }$-finite.
${\displaystyle \operatorname {BV} ([a,b])}$	?	No	Yes	${\displaystyle \|f\|_{BV}}$	${\displaystyle =V_{f}([a,b])+\lim \nolimits _{x\to a^{+}}f(x)}$	${\displaystyle V_{f}([a,b]).}$ is the total variation of ${\displaystyle f}$
${\displaystyle \operatorname {NBV} ([a,b])}$	?	No	Yes	${\displaystyle \|f\|_{BV}}$	${\displaystyle =V_{f}([a,b])}$	${\displaystyle \operatorname {NBV} ([a,b])}$ consists of ${\displaystyle \operatorname {BV} ([a,b])}$ functions such that ${\displaystyle \lim \nolimits _{x\to a^{+}}f(x)=0}$
${\displaystyle \operatorname {AC} ([a,b])}$	${\displaystyle \mathbb {F} +L^{\infty }([a,b])}$	No	Yes	${\displaystyle \|f\|_{BV}}$	${\displaystyle =V_{f}([a,b])+\lim \nolimits _{x\to a^{+}}f(x)}$	Isomorphic to the Sobolev space ${\displaystyle W^{1,1}([a,b]).}$
${\displaystyle C^{n}([a,b])}$	${\displaystyle \operatorname {rca} ([a,b])}$	No	No	${\displaystyle \|f\|}$	${\displaystyle =\sum _{i=0}^{n}\sup \nolimits _{x\in [a,b]}\left|f^{(i)}(x)\right|}$	Isomorphic to ${\displaystyle \mathbb {R} ^{n}\oplus C([a,b]),}$ essentially by Taylor's theorem.

• The Asplund spaces
• The Hardy spaces
• The space ${\displaystyle \operatorname {BMO} }$ of functions of bounded mean oscillation
• The space of functions of bounded variation
• Sobolev spaces
• The Birnbaum–Orlicz spaces ${\displaystyle L^{A}(\mu ).}$
• Hölder spaces ${\displaystyle C^{k}(\Omega ).}$
• Lorentz space

• James' space, a Banach space that has a Schauder basis, but has no unconditional Schauder Basis. Also, James' space is isometrically isomorphic to its double dual, but fails to be reflexive.
• Tsirelson space, a reflexive Banach space in which neither ${\displaystyle \ell ^{p}}$ nor ${\displaystyle c_{0}}$ can be embedded.
• W.T. Gowers construction of a space ${\displaystyle X}$ that is isomorphic to ${\displaystyle X\oplus X\oplus X}$ but not ${\displaystyle X\oplus X}$ serves as a counterexample for weakening the premises of the Schroeder–Bernstein theorem[1]

• List of mathematical spaces – Mathematical set with some added structurePages displaying short descriptions of redirect targets
• List of topologies – List of concrete topologies and topological spaces
• Minkowski distance – Mathematical metric in normed vector space

• 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.