View source for List of Banach spaces

In the [[Mathematics|mathematical]] field of [[Functional analysis|functional analysis]], [[Banach space]]s are among the most important objects of study.  In other areas of [[Mathematical analysis|mathematical analysis]], most spaces which arise in practice turn out to be Banach spaces as well.
== 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.

{{clear}}

== Banach spaces in other areas of analysis ==

* The [[Asplund space]]s
* The [[Hardy space]]s
* The space <math>\operatorname{BMO}</math> of functions of [[Bounded mean oscillation|bounded mean oscillation]]
* The space of functions of [[Bounded variation|bounded variation]]
* [[Sobolev space]]s
* The [[Birnbaum–Orlicz space]]s <math>L^A(\mu).</math>
* Hölder spaces <math>C^k(\Omega).</math>
* [[Lorentz space]]

== Banach spaces serving as counterexamples ==

* [[James' space]], a Banach space that has a [[Schauder basis]], but has no [[Schauder basis#Unconditionality|unconditional Schauder Basis]]. Also, James' space is isometrically isomorphic to its double dual, but fails to be reflexive.
* [[Tsirelson space]], a reflexive Banach space in which neither [[Lp space|<math>\ell^p</math>]] nor [[Sequence space#c and c0|<math>c 0</math>]] can be embedded.
* W.T. Gowers construction of a space <math>X</math> 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 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>

== See also ==
* {{annotated link|List of mathematical spaces}}
* {{annotated link|List of topologies}}
* {{annotated link|Minkowski distance}}

== Notes ==

{{reflist|group=note}}
{{reflist}}

==References==

{{reflist}}

* {{citation|title=Sequences and series in Banach spaces|first=Joseph|last=Diestel|publisher=Springer-Verlag|year=1984|isbn=0-387-90859-5|url-access=registration|url=https://archive.org/details/sequencesseriesi0000dies}}.
* {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|title=Linear operators, Part I|publisher=Wiley-Interscience|year=1958}}.

{{Functional Analysis}}

{{DEFAULTSORT:Banach spaces}}
[[Category:Banach spaces]]
[[Category:Mathematics-related lists]]

{{Sourceattribution|List of Banach spaces}}