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.

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 (also with 1 < q < ∞), so that the next equation holds:

${\displaystyle {\frac {1}{q}}+{\frac {1}{p}}=1,}$

and thus

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

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	${\displaystyle \|x\|_{2}=\left(\sum _{i=1}^{n}|x_{i}|^{2}\right)^{1/2}}$
ℓnp	ℓnq	Yes	Yes	${\displaystyle \|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{1/p}}$
ℓn∞	ℓn1	Yes	Yes	${\displaystyle \|x\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|}$
ℓp	ℓq	Yes	Yes	${\displaystyle \|x\|_{p}=\left(\sum _{i=1}^{\infty }|x_{i}|^{p}\right)^{1/p}}$	1 < p < ∞
ℓ1	ℓ∞	No	Yes	${\displaystyle \|x\|_{1}=\sum _{i=1}^{\infty }|x_{i}|}$
ℓ∞	ba	No	No	${\displaystyle \|x\|_{\infty }=\sup _{i}|x_{i}|}$
c	ℓ1	No	No	${\displaystyle \|x\|_{\infty }=\sup _{i}|x_{i}|}$
c0	ℓ1	No	No	${\displaystyle \|x\|_{\infty }=\sup _{i}|x_{i}|}$	Isomorphic but not isometric to c.
bv	${\displaystyle \ell _{\infty }}$	No	Yes	${\displaystyle \|x\|_{bv}=|x_{1}|+\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|}$	isomorphic to ${\displaystyle \ell _{1}}$
bv0	${\displaystyle \ell _{\infty }}$	No	Yes	${\displaystyle \|x\|_{bv_{0}}=\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|}$	isometrically isomorphic to ${\displaystyle \ell _{1}}$
bs	ba	No	No	${\displaystyle \|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|}$	Isometrically isomorphic to ℓ∞.
cs	ℓ1	No	No	${\displaystyle \|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|}$	Isometrically isomorphic to c.
B(X, Ξ)	ba(Ξ)	No	No	${\displaystyle \|f\|_{B}=\sup _{x\in X}|f(x)|}$
C(X)	rca(X)	No	No	${\displaystyle \|f\|_{B}=\sup _{x\in X}\left|f(x)\right|}$	X is a compact Hausdorff space.
ba(Ξ)	?	No	Yes	${\displaystyle \|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)}$ (variation of a measure)
ca(Σ)	?	No	Yes	${\displaystyle \|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)}$
rca(Σ)	?	No	Yes	${\displaystyle \|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)}$
Lp(μ)	Lq(μ)	Yes	Yes	${\displaystyle \|f\|_{p}=\left\{\int |f|^{p}\,d\mu \right\}^{1/p}}$	1 < p < ∞
L1(μ)	L∞(μ)	No	?	${\displaystyle \|f\|_{1}=\int |f|\,d\mu }$	If the measure μ on S is sigma-finite
L∞(μ)	${\displaystyle N_{\mu }^{\perp }}$	No	?	${\displaystyle \|f\|_{\infty }\equiv \inf\{C\geq 0:|f(x)|\leq C{\mbox{ for almost every }}x\}.}$	where ${\displaystyle N_{\mu }^{\perp }=\{\sigma \in ba(\Sigma ):\lambda \ll \mu \}}$
BV(I)	?	No	Yes	${\displaystyle \|f\|_{BV}=\lim _{x\to a^{+}}|f(x)|+V_{f}(I)}$	Vf(I) is the total variation of f.
NBV(I)	?	No	Yes	${\displaystyle \|f\|_{BV}=V_{f}(I)}$	NBV(I) consists of BV functions such that ${\displaystyle \lim _{x\to a^{+}}f(x)=0}$.
AC(I)	K+L∞(I)	No	Yes	${\displaystyle \|f\|_{BV}=\lim _{x\to a^{+}}|f(x)|+V_{f}(I)}$	Isomorphic to the Sobolev space W1,1(I).
Cn[a,b]	rca([a,b])	No	No	${\displaystyle \|f\|=\sum _{i=0}^{n}\sup _{x\in [a,b]}|f^{(i)}(x)|.}$	Isomorphic to Rn ⊕ C([a,b]), essentially by Taylor's theorem.

• The Asplund spaces
• 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(μ).
• Hölder spaces C^k,α(Ω).
• 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 ℓ^p nor c₀ can be embedded.
• W.T. Gowers construction of a space 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]

• 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.