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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.

	Dual space	Reflexive	weakly sequentially complete	Norm		Notes
Classical Banach spaces
$\mathbb {F} ^{n}$	$\mathbb {F} ^{n}$	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	$\\|x\\|_{bv}$	$=\left\|x_{1}\right\|+\sum _{i=1}^{\infty }\left\|x_{i+1}-x_{i}\right\|$	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	$\\|\mu \\|_{ba}$	$=\sup \nolimits _{S\in \Sigma }\|\mu \|(S)$
$\operatorname {ca} (\Sigma )$	?	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 $\operatorname {ca} (\Sigma ).$
$L^{p}(\mu )$	$L^{q}(\mu )$	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.

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