Template:Common Banach spaces

Glossary of symbols for the table below:

• $𝔽$ denotes the field of real numbers $ℝ$ or complex numbers $ℂ.$
• $K$ is a compact Hausdorff space.
• $p,q\in ℝ$ are real numbers with $1<p,q<\mathrm{\infty }$ that are Hölder conjugates, meaning that they satisfy $\frac{1}{q}+\frac{1}{p}=1$ and thus also $q=\frac{p}{p-1}.$
• $\mathrm{\Sigma }$ is a $\sigma$-algebra of sets.
• $\mathrm{\Xi }$ 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
${𝔽}^n$	${𝔽}^n$	Template:Yes	Template:Yes	$\Vert x{\Vert }_2$	$={(\sum _{i=1}^n|x_i{|}^2)}^{1/2}$	Euclidean space
${\ell }_p^n$	${\ell }_q^n$	Template:Yes	Template:Yes	$\Vert x{\Vert }_p$	$={(\sum _{i=1}^n|x_i{|}^p)}^{\frac{1}{p}}$
${\ell }_{\mathrm{\infty }}^n$	${\ell }_1^n$	Template:Yes	Template:Yes	$\Vert x{\Vert }_{\mathrm{\infty }}$	$=\max {\mathrm{\backslash nolimits}}_{1\le i\le n}|x_i|$
${\ell }^p$	${\ell }^q$	Template:Yes	Template:Yes	$\Vert x{\Vert }_p$	$={(\sum _{i=1}^{\mathrm{\infty }}|x_i{|}^p)}^{\frac{1}{p}}$
${\ell }^1$	${\ell }^{\mathrm{\infty }}$	Template:No	Template:Yes	$\Vert x{\Vert }_1$	$=\sum _{i=1}^{\mathrm{\infty }}\left|x_i\right|$
${\ell }^{\mathrm{\infty }}$	$\mathrm{ba}$	Template:No	Template:No	$\Vert x{\Vert }_{\mathrm{\infty }}$	$=\sup {\mathrm{\backslash nolimits}}_i\left|x_i\right|$
$\mathrm{c}$	${\ell }^1$	Template:No	Template:No	$\Vert x{\Vert }_{\mathrm{\infty }}$	$=\sup {\mathrm{\backslash nolimits}}_i\left|x_i\right|$
$c_0$	${\ell }^1$	Template:No	Template:No	$\Vert x{\Vert }_{\mathrm{\infty }}$	$=\sup {\mathrm{\backslash nolimits}}_i\left|x_i\right|$	Isomorphic but not isometric to $c.$
$\mathrm{bv}$	${\ell }^{\mathrm{\infty }}$	Template:No	Template:Yes	$\Vert x{\Vert }_{bv}$	$=\left|x_1\right|+\sum _{i=1}^{\mathrm{\infty }}|x_{i+1}-x_i|$	Isometrically isomorphic to ${\ell }^1.$
${\mathrm{bv}}_0$	${\ell }^{\mathrm{\infty }}$	Template:No	Template:Yes	$\Vert x{\Vert }_{b{v}_0}$	$=\sum _{i=1}^{\mathrm{\infty }}|x_{i+1}-x_i|$	Isometrically isomorphic to ${\ell }^1.$
$\mathrm{bs}$	$\mathrm{ba}$	Template:No	Template:No	$\Vert x{\Vert }_{bs}$	$=\sup {\mathrm{\backslash nolimits}}_n\left|\sum _{i=1}^n{x}_i\right|$	Isometrically isomorphic to ${\ell }^{\mathrm{\infty }}.$
$\mathrm{cs}$	${\ell }^1$	Template:No	Template:No	$\Vert x{\Vert }_{bs}$	$=\sup {\mathrm{\backslash nolimits}}_n\left|\sum _{i=1}^n{x}_i\right|$	Isometrically isomorphic to $c.$
$B(K,\mathrm{\Xi })$	$\mathrm{ba}(\mathrm{\Xi })$	Template:No	Template:No	$\Vert f{\Vert }_B$	$=\sup {\mathrm{\backslash nolimits}}_{k\in K}|f(k)|$
$C(K)$	$\mathrm{rca}(K)$	Template:No	Template:No	$\Vert x{\Vert }_{C(K)}$	$=\max {\mathrm{\backslash nolimits}}_{k\in K}|f(k)|$
$\mathrm{ba}(\mathrm{\Xi })$	?	Template:No	Template:Yes	$\Vert \mu {\Vert }_{ba}$	$=\sup {\mathrm{\backslash nolimits}}_{S\in \mathrm{\Xi }}|\mu |(S)$
$\mathrm{ca}(\mathrm{\Sigma })$	?	Template:No	Template:Yes	$\Vert \mu {\Vert }_{ba}$	$=\sup {\mathrm{\backslash nolimits}}_{S\in \mathrm{\Sigma }}|\mu |(S)$	A closed subspace of $\mathrm{ba}(\mathrm{\Sigma }).$
$\mathrm{rca}(\mathrm{\Sigma })$	?	Template:No	Template:Yes	$\Vert \mu {\Vert }_{ba}$	$=\sup {\mathrm{\backslash nolimits}}_{S\in \mathrm{\Sigma }}|\mu |(S)$	A closed subspace of $\mathrm{ca}(\mathrm{\Sigma }).$
$L^p(\mu )$	$L^q(\mu )$	Template:Yes	Template:Yes	$\Vert f{\Vert }_p$	$={(\int |f{|}^{p}d\mu )}^{\frac{1}{p}}$
$L^1(\mu )$	$L^{\mathrm{\infty }}(\mu )$	Template:No	Template:Yes	$\Vert f{\Vert }_1$	$=\int |f|d\mu$	The dual is $L^{\mathrm{\infty }}(\mu )$ if $\mu$ is $\sigma$-finite.
$\mathrm{BV}([a,b])$	?	Template:No	Template:Yes	$\Vert f{\Vert }_{BV}$	$=V_f([a,b])+\lim {\mathrm{\backslash nolimits}}_{x\to a^{+}}f(x)$	$V_f([a,b])$ is the total variation of $f$
$\mathrm{NBV}([a,b])$	?	Template:No	Template:Yes	$\Vert f{\Vert }_{BV}$	$=V_f([a,b])$	$\mathrm{NBV}([a,b])$ consists of $\mathrm{BV}([a,b])$ functions such that $\lim {\mathrm{\backslash nolimits}}_{x\to a^{+}}f(x)=0$
$\mathrm{AC}([a,b])$	$𝔽+L^{\mathrm{\infty }}([a,b])$	Template:No	Template:Yes	$\Vert f{\Vert }_{BV}$	$=V_f([a,b])+\lim {\mathrm{\backslash nolimits}}_{x\to a^{+}}f(x)$	Isomorphic to the Sobolev space $W^{1,1}([a,b]).$
$C^n([a,b])$	$\mathrm{rca}([a,b])$	Template:No	Template:No	$\Vert f\Vert$	$=\sum _{i=0}^n\sup {\mathrm{\backslash nolimits}}_{x\in [a,b]}|f^{(i)}(x)|$	Isomorphic to ${ℝ}^n\oplus C([a,b]),$ essentially by Taylor's theorem.