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In [[functional analysis]] and related areas of [[mathematics]], '''Smith space''' is a [[Uniform space|complete]] [[compactly generated space|compactly generated]] [[locally convex space]] <math>X</math> having a compact set <math>K</math> which absorbs every other compact set <math>T\subseteq X</math> (i.e. <math>T\subseteq\lambda\cdot K</math> for some <math>\lambda>0</math>).

Smith spaces are named after M. F. Smith,<ref name=Smith>{{harvtxt|M. F. Smith|1952}}.</ref> who introduced them as duals to [[Banach space]]s in some versions of duality theory for [[topological vector space]]s. All Smith spaces are [[stereotype space|stereotype]] and are in the stereotype duality relations with [[Banach space]]s:<ref name=Akbarov-1>{{harvtxt|S.S.Akbarov|2003}}.</ref><ref name=Akbarov-2>{{harvtxt|S.S.Akbarov|2009}}.</ref>
:* for any Banach space <math>X</math> its stereotype dual space<ref>The ''stereotype dual'' space to a locally convex space <math>X</math> is the space <math>X^\star</math> of all linear continuous functionals <math>f:X\to\mathbb{C}</math> endowed with the topology of uniform convergence on [[totally bounded set]]s in <math>X</math>.</ref> <math>X^\star</math> is a Smith space,

:* and vice versa, for any Smith space <math>X</math> its stereotype dual space <math>X^\star</math> is a Banach space.

== Notes ==

{{reflist}}

== References ==

* {{cite book
 | last = Schaefer
 | first = Helmuth H. <!-- | authorlink = Helmuth Schaefer -->
 | year = 1966
 | title = Topological vector spaces
 | series=
 | volume=
 | publisher = The MacMillan Company
 | location = New York
 | isbn = 0-387-98726-6
}}

* {{cite book |last=Robertson |first=A.P. |coauthors=Robertson, W.J.|title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] }}
*{{cite journal|last=Smith|first=M.F.|title=[http://www.jstor.org/stable/1969798 The Pontrjagin duality theorem in linear spaces]|journal=Annals of Mathematics|year=1952|volume=56|issue=2|pages=248–253|doi=10.2307/1969798}}

*{{cite journal|last=Akbarov|first=S.S.|title=[http://www.springerlink.com/content/k62m72960101g6q2/ Pontryagin duality in the theory of topological vector spaces and in topological algebra]|journal=Journal of Mathematical Sciences|year=2003|volume=113|issue=2|pages=179–349|doi=10.1023/A:1020929201133}}

*{{cite journal|last=Akbarov|first=S.S.|title=[http://www.springerlink.com/content/u07317731010573l/ Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity]{{subscription required}}|journal=Journal of Mathematical Sciences|year=2009|volume=162|issue=4|pages=459–586|doi=10.1007/s10958-009-9646-1}}

{{Functional Analysis}}

[[Category:Functional analysis]]

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