Template:Common Banach spaces

Glossary of symbols for the table below:

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

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