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Glossary of symbols for the table below:
* <math>\mathbb{F}</math> denotes the [[Field (mathematics)|field]] of [[real numbers]] <math>\R</math> or [[complex numbers]] <math>\Complex.</math>
* <math>K</math> is a [[compact Hausdorff space]].
* <math>p, q \in \R</math> are [[real number]]s with <math>1 < p, q < \infty</math> that are [[Hölder conjugate]]s, meaning that they satisfy <math>\frac{1}{q} + \frac{1}{p} = 1</math> and thus also <math>q = \frac{p}{p-1}.</math>
* <math>\Sigma</math> is a [[Sigma algebra|<math>\sigma</math>-algebra]] of sets.
* <math>\Xi</math> is an algebra of sets (for spaces only requiring finite additivity, such as the [[ba space]]).
* <math>\mu</math> is a [[Measure (mathematics)|measure]] with [[Total variation#Total variation in measure theory|variation]] <math>|\mu|.</math> A positive measure is a real-valued positive set function defined on a <math>\sigma</math>-algebra which is countably additive.

{| align="left" class="wikitable" style="text-align:center"
|- align="center"
| style="border-bottom: 2px solid #303060" colspan=7| '''Classical [[Banach space]]s'''
|-
! !! [[Dual space]] !! [[Reflexive space|Reflexive]] !! [[Banach space#Weak convergences of sequences|weakly sequentially complete]] !! colspan="2"|[[Normed space|Norm]] !! Notes
|-
! [[Euclidean space|<math>\mathbb{F}^n</math>]]
|<math>\mathbb{F}^n</math> ||{{yes}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_2</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \left(\sum_{i=1}^n |x_i|^2\right)^{1/2}</math>
|Euclidean space
|-
! [[Lp space|<math>\ell^n_p</math>]]
|<math>\ell^n_q</math> || {{yes}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_p</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \left(\sum_{i=1}^n |x_i|^p\right)^{\frac{1}{p}}</math>
|
|-
! [[Lp space|<math>\ell^n_{\infty}</math>]]
|<math>\ell^n_1</math>|| {{yes}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|[[Uniform norm|<math>\|x\|_\infty </math>]]
|style="text-align: left; border-style: solid none solid none;"|<math>= \max\nolimits_{1\le i\le n} |x_i|</math>
|
|-
! [[Lp space|<math>\ell^p</math>]]
|<math>\ell^q</math> || {{yes}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_p</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \left(\sum_{i=1}^\infty |x_i|^p\right)^{\frac{1}{p}}</math>
|
|-
! [[Lp space|<math>\ell^1</math>]]
|<math>\ell^{\infty}</math> || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_1</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sum_{i=1}^\infty \left|x_i\right|</math>
|
|-
! [[L-infinity|<math>\ell^{\infty}</math>]]
|[[ba space|<math>\operatorname{ba}</math>]] || {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|[[Uniform norm|<math>\|x\|_\infty</math>]]
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_i \left|x_i\right|</math>
|
|-
! [[c space|<math>\operatorname{c}</math>]]
|<math>\ell^1</math> || {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|[[Uniform norm|<math>\|x\|_\infty</math>]]
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_i \left|x_i\right|</math>
|
|-
! [[Space of null sequences|<math>c_0</math>]]
|<math>\ell^1</math> || {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|[[Uniform norm|<math>\|x\|_\infty</math>]]
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_i \left|x_i\right|</math>
|Isomorphic but not isometric to <math>c.</math>
|-
! [[bv space|<math>\operatorname{bv}</math>]]
|<math>\ell^{\infty}</math> || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_{bv}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \left|x_1\right| + \sum_{i=1}^\infty \left|x_{i+1} - x_i\right|</math>
|Isometrically isomorphic to <math>\ell^1.</math>
|-
! [[bv space|<math>\operatorname{bv}_0</math>]]
|<math>\ell^{\infty}</math> || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_{bv_0}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sum_{i=1}^\infty \left|x_{i+1} - x_i\right|</math>
|Isometrically isomorphic to <math>\ell^1.</math>
|-
! [[bs space|<math>\operatorname{bs}</math>]]
|[[ba space|<math>\operatorname{ba}</math>]] || {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_{bs}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_n \left|\sum_{i=1}^n x_i\right|</math>
|Isometrically isomorphic to <math>\ell^{\infty}.</math>
|-
! [[cs space|<math>\operatorname{cs}</math>]]
|<math>\ell^1</math>|| {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_{bs}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_n \left|\sum_{i=1}^nx_i\right|</math>
|Isometrically isomorphic to <math>c.</math>
|-
! [[ba space|<math>B(K, \Xi)</math>]]
|[[ba space|<math>\operatorname{ba}(\Xi)</math>]] || {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|f\|_B</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_{k \in K} |f(k)|</math>
|
|-
! [[Continuous functions on a compact Hausdorff space|<math>C(K)</math>]]
|[[ba space|<math>\operatorname{rca}(K)</math>]] || {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|x\|_{C(K)}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \max\nolimits_{k \in K} |f(k)|</math>
|
|-
! [[ba space|<math>\operatorname{ba}(\Xi)</math>]]
| ? || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|\mu\|_{ba}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_{S\in\Sigma} |\mu|(S)</math>
|
|-
! [[ba space|<math>\operatorname{ca}(\Sigma)</math>]]
| ? || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|\mu\|_{ba}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_{S\in\Sigma} |\mu|(S)</math>
|A closed subspace of <math>\operatorname{ba}(\Sigma).</math>
|-
! [[ba space|<math>\operatorname{rca}(\Sigma)</math>]]
| ? || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|\mu\|_{ba}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sup\nolimits_{S\in\Sigma} |\mu|(S)</math>
|A closed subspace of <math>\operatorname{ca}(\Sigma).</math>
|-
! [[Lp space|<math>L^p(\mu)</math>]]
|<math>L^q(\mu)</math> || {{yes}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|f\|_p</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \left (\int |f|^p\,d\mu\right)^{\frac{1}{p}}</math>
|
|-
! [[L1 space|<math>L^1(\mu)</math>]]
|<math>L^{\infty}(\mu)</math> || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|f\|_1</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \int |f|\,d\mu</math>
|The dual is <math>L^{\infty}(\mu)</math> if <math>\mu</math> is [[σ-finite measure|<math>\sigma</math>-finite]].
|-
! [[Bounded variation|<math>\operatorname{BV}([a, b])</math>]]
| ? || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|f\|_{BV}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= V_f([a, b]) + \lim\nolimits_{x \to a^+}f(x)</math>
|<math>V_f([a, b]).</math> is the [[total variation]] of <math>f</math>
|-
! [[Bounded variation|<math>\operatorname{NBV}([a, b])</math>]]
| ? || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|f\|_{BV}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= V_f([a, b])</math>
|<math>\operatorname{NBV}([a, b])</math> consists of <math>\operatorname{BV}([a, b])</math> functions such that <math>\lim\nolimits_{x \to a^+} f(x) = 0</math>
|-
! [[Absolutely continuous function|<math>\operatorname{AC}([a, b])</math>]]
|<math>\mathbb{F} + L^{\infty}([a, b])</math> || {{no}} || {{yes}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|f\|_{BV}</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= V_f([a, b]) + \lim\nolimits_{x \to a^+} f(x)</math>
|Isomorphic to the [[Sobolev space]] <math>W^{1, 1}([a, b]).</math>
|-
! [[Continuously differentiable|<math>C^n([a, b])</math>]]
|[[Ba space|<math>\operatorname{rca}([a, b])</math>]] || {{no}} || {{no}}
|style="text-align: left; border-style: solid none solid none;"|<math>\|f\|</math>
|style="text-align: left; border-style: solid none solid none;"|<math>= \sum_{i=0}^n \sup\nolimits_{x \in [a,b]} \left|f^{(i)}(x)\right|</math>
|Isomorphic to <math>\R^n \oplus C([a, b]),</math> essentially by [[Taylor's theorem]].
|}
<noinclude>
[[Category:Banach spaces| ]]
[[Category:Mathematics navigational boxes]]
[[Category:Normed spaces| ]]
</noinclude>

Template:Common Banach spaces - Revision history

Indicwiki: 1 revision imported from :alpha:Template:Common_Banach_spaces

alpha>Indicwiki: Created page with "Glossary of symbols for the table below: * \mathbb{F} denotes the field of real numbers \R or complex numbers \..."

alpha>Indicwiki: Created page with "Glossary of symbols for the table below: * $\mathbb{F}$ denotes the field of real numbers $\R$ or complex numbers $\..."$