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	$\|x\|_{\infty }$	$=\sup \nolimits _{i}\left|x_{i}\right|$
$\operatorname {c}$	$\ell ^{1}$	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	${\Vert <!--\; \Vert \; -->x\Vert <!--\; \Vert \; -->}_{}$