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In the [[mathematics|mathematical]] field of [[functional analysis]], [[Banach space]]s are among the most important objects of study. In other areas of [[mathematical analysis]], most spaces which arise in practice turn out to be Banach spaces as well.
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== Classical Banach spaces ==
According to {{harvtxt|Diestel|1984|loc=Chapter VII}}, the '''classical Banach spaces''' are those defined by {{harvtxt|Dunford|Schwartz|1958}}, which is the source for the following table.
 
Here '''K''' denotes the [[field (mathematics)|field]] of [[real numbers]] or [[complex numbers]] and ''I'' is a closed and bounded interval [''a'',''b''].  The number ''p'' is a [[real number]] with {{nowrap|1 < ''p'' < ∞}}, and ''q'' is its [[Hölder conjugate]] (also with {{nowrap|1 < ''q'' < ∞}}), so that the next equation holds:
 
: <math> \frac{1}{q}+\frac{1}{p}=1 ,</math>  
 
and thus
 
: <math> q=\frac{p}{p-1} .</math>  
 
The symbol &Sigma; denotes a [[sigma algebra|&sigma;-algebra]] of sets, and &Xi; denotes just an algebra of sets (for spaces only requiring finite additivity, such as the [[ba space]]).  The symbol &mu; denotes a positive measure: that is, a real-valued positive set function defined on a &sigma;-algebra which is countably additive.
 
{| align="left" class="wikitable" style="text-align:center"
|- align="center"
| style="border-bottom: 2px solid #303060" colspan=6| '''Classical Banach spaces'''
|-
! !! [[Dual space]] !! [[Reflexive space|Reflexive]] !! [[Weak topology|weakly]]  [[complete metric space|complete]] !! [[Normed space|Norm]] !! Notes
|-
! [[Euclidean space|'''K'''<sup>n</sup>]]
| '''K'''<sup>n</sup> ||{{yes}} || {{yes}} || <math>\|x\|_2 = \left(\sum_{i=1}^n |x_i|^2\right)^{1/2}</math> ||
|-
!  [[Lp space|ℓ<sup>n</sup><sub>p</sub>]] ||  ℓ<sup>n</sup><sub>q</sub> || {{yes}} || {{yes}} || <math>\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}</math> ||
|-
! [[Lp space|ℓ<sup>n</sup><sub>∞</sub>]]
| ℓ<sup>n</sup><sub>1</sub> || {{yes}} || {{yes}} || <math>\|x\|_\infty = \max_{1\le i\le n} |x_i|</math> ||
|-
!  [[Lp space|ℓ<sub>p</sub>]] ||  ℓ<sub>q</sub> || {{yes}} || {{yes}} || <math>\|x\|_p = \left(\sum_{i=1}^\infty |x_i|^p\right)^{1/p}</math> || 1 < p < ∞
|-
!  [[Lp space|ℓ<sub>1</sub>]] ||  ℓ<sub>∞</sub> || {{no}} || {{yes}} || <math>\|x\|_1 = \sum_{i=1}^\infty |x_i|</math> ||
|-
!  [[Lp space|ℓ<sub>∞</sub>]] ||  [[ba space|ba]] || {{no}} || {{no}} || <math>\|x\|_\infty = \sup_i |x_i|</math> ||
|-
! [[c space|''c'']]
| ℓ<sub>1</sub> || {{no}} || {{no}} || <math>\|x\|_\infty = \sup_i |x_i|</math> ||
|-
! [[c0 space|''c''<sub>0</sub>]]
| ℓ<sub>1</sub> || {{no}} || {{no}} || <math>\|x\|_\infty = \sup_i |x_i|</math> || Isomorphic but not isometric to ''c''.
|-
! [[bv space|''bv'']]
| ℓ<sub>1</sub> + '''K''' || {{no}} || {{yes}} || <math>\|x\|_{bv} = |x_1| + \sum_{i=1}^\infty|x_{i+1}-x_i|</math> ||
|-
! [[bv space|''bv''<sub>0</sub>]]
| ℓ<sub>1</sub> || {{no}} || {{yes}} || <math>\|x\|_{bv_0} = \sum_{i=1}^\infty|x_{i+1}-x_i|</math> ||
|-
! [[bs space|''bs'']]
| [[ba space|ba]] || {{no}} || {{no}} || <math>\|x\|_{bs} = \sup_n\left|\sum_{i=1}^nx_i\right|</math> || Isometrically isomorphic to ℓ<sub>∞</sub>.
|-
! [[bs space|''cs'']]
| ℓ<sub>1</sub> || {{no}} || {{no}} || <math>\|x\|_{bs} = \sup_n\left|\sum_{i=1}^nx_i\right|</math> || Isometrically isomorphic to [[c space|c]].
|-
! [[ba space|''B''(''X'',&nbsp;&Xi;)]] || [[ba space|ba(&Xi;)]] || {{no}} || {{no}} ||  <math>\|f\|_B = \sup_{x\in X}|f(x)|</math> ||
|-
! [[Continuous functions on a compact Hausdorff space|''C''(''X'')]]
| [[ba space|''rca''(''X'')]] || {{no}} || {{no}} || <math>\|f\|_{B} = \sup_{x\in X}\left|f(x)\right|</math> || ''X'' is a [[compact Hausdorff space]].
|-
! [[ba space|ba(&Xi;)]]
| ? || {{no}} || {{yes}} || <math>\|\mu\|_{ba} = \sup_{A\in\Sigma} |\mu|(A)</math>
([[Total_variation#Total_variation_in_measure_theory|variation of a measure]])
||
|-
! [[ba space|ca(&Sigma;)]]
| ? || {{no}} || {{yes}} || <math>\|\mu\|_{ba} = \sup_{A\in\Sigma} |\mu|(A)</math> ||
|-
! [[ba space|rca(&Sigma;)]]
| ? || {{no}} || {{yes}} || <math>\|\mu\|_{ba} = \sup_{A\in\Sigma} |\mu|(A)</math> ||
|-
! [[Lp space|L<sup>p</sup>(&mu;)]]
| L<sup>q</sup>(&mu;) || {{yes}} || {{yes}} || <math>\|f\|_p = \left\{\int |f|^p\,d\mu\right\}^{1/p}</math> || 1 < p < ∞
|-
! [[Lp space|L<sup>1</sup>(&mu;)]]
| L<sup>∞</sup>(&mu;) || {{no}} || ? || <math>\|f\|_1 = \int |f|\,d\mu</math> || If the measure ''μ'' on ''S'' is [[sigma-finite]]
|-
! [[Lp space|L<sup>∞</sup>(&mu;)]]
| [[ba space|<math>N_\mu^\perp</math>]] || {{no}} || ? || <math>\|f\|_\infty \equiv \inf \{ C\ge 0 : |f(x)| \le C \mbox{ for almost every } x\}.</math> || where <math>N_\mu^\perp =\{\sigma\in ba(\Sigma) : \lambda \ll \mu\}</math>
|-
! [[Bounded variation|BV(I)]]
| ? || {{no}} || {{yes}} || <math>\|f\|_{BV} = \lim_{x\to a^+}f(x) + V_f(I)</math> || ''V''<sub>f</sub>(''I'') is the [[total variation]] of ''f''.
|-
! [[Bounded variation|NBV(I)]]
| ? || {{no}} || {{yes}} || <math>\|f\|_{BV} = V_f(I)</math> || NBV(''I'') consists of BV functions such that <math>\lim_{x\to a^+}f(x)=0</math>.
|-
! [[Absolutely continuous function|AC(I)]]
| '''K'''+''L''<sup>∞</sup>(''I'') || {{no}} || {{yes}} || <math>\|f\|_{BV} = \lim_{x\to a^+}f(x) + V_f(I)</math> || Isomorphic to the [[Sobolev space]] ''W''<sup>1,1</sup>(''I'').
|-
! [[Continuously differentiable|C<sup>''n''</sup>[''a'',''b'']]]
|| [[Ba space|rca([''a'',''b''])]] || {{no}} || {{no}} || <math>\|f\| = \sum_{i=0}^n \sup_{x\in [a,b]} |f^{(i)}(x)|.</math> || Isomorphic to '''R'''<sup>''n''</sup>&nbsp;&oplus;&nbsp;C([''a'',''b'']), essentially by [[Taylor's theorem]].
|}
 
{{clr}}
 
==Banach spaces in other areas of analysis==
* The [[Asplund space|Asplund spaces]]
* The [[Hardy space]]s
* The space ''BMO'' of functions of [[bounded mean oscillation]]
* The space of functions of [[bounded variation]]
* [[Sobolev space]]s
* The [[Birnbaum–Orlicz space]]s ''L''<sub>A</sub>(&mu;).
* [[Hölder space]]s C<sup>k,&alpha;</sup>(&Omega;).
* [[Lorentz space]]
 
==Banach spaces serving as counterexamples==
* [[Tsirelson space]]
* [[W.T. Gowers]] construction of a space ''X'' that is isomorphic to <math>X\oplus X\oplus X</math> but not <math>X\oplus X</math> serves as a counterexample for weakening the premises of the [[Cantor–Bernstein–Schroeder theorem|Schroeder–Bernstein theorem]] <ref>W.T. Gowers, "A solution to the Schroeder–Bernstein problem for Banach spaces", ''Bulletin of the London Mathematical Society'', '''28''' (1996) pp. 297–304.</ref>
 
==Notes==
<references/>
 
==References==
* {{citation|title=Sequences and series in Banach spaces|first=Joseph|last=Diestel|publisher=Springer-Verlag|year=1984|isbn=0-387-90859-5}}.
* {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|title=Linear operators, Part I|publisher=Wiley-Interscience|year=1958}}.
 
[[Category:Functional analysis]]
[[Category:Banach spaces]]

Latest revision as of 00:20, 2 March 2014

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