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In [[Functional analysis|functional analysis]] and related areas of [[Mathematics|mathematics]], a '''Smith space''' is a [[Uniform space|complete]] [[Compactly generated space|compactly generated]] [[Locally convex topological vector space|locally convex topological vector space]] <math>X</math> having a ''universal compact set'', i.e. 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 
[https://www.genealogy.math.ndsu.nodak.edu/id.php?id=4032 Marianne Ruth Freundlich Smith], who introduced them{{sfn|Smith|1952}} as duals to [[Banach space]]s in some versions of duality theory for [[Topological vector space|topological vector space]]s. All Smith spaces are [[Stereotype space|stereotype]] and are in the stereotype duality relations with [[Banach space]]s:{{sfn|Akbarov|2003|p=220}}{{sfn|Akbarov|2009|p=467}}
:* 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 sets 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.

Smith spaces are special cases of [[Brauner space]]s.

==Examples==

* As follows from the duality theorems, for any Banach space <math>X</math> its stereotype dual space <math>X^\star</math> is a Smith space. The [[Polar set|polar]] <math>K=B^\circ</math> of the unit ball <math>B</math> in <math>X</math> is the universal compact set in <math>X^\star</math>. If <math>X^*</math> denotes the [[Banach space#Dual space|normed dual space]] for <math>X</math>, and <math>X'</math> the space <math>X^*</math> endowed with the <math>X</math>-weak topology, then the topology of <math>X^\star</math> lies between the topology of <math>X^*</math> and the topology of <math>X'</math>, so there are natural (linear continuous) bijections
:: <math>X^*\to X^\star\to X'.</math>
: If <math>X</math> is infinite-dimensional, then no two of these topologies coincide. At the same time, for infinite dimensional <math>X</math> the space <math>X^\star</math> is not barreled (and even is not a [[Mackey space]] if <math>X</math> is [[Reflexive space#Reflexive Banach spaces|reflexive as a Banach space]]{{sfn|Akbarov|2003|loc=Example 4.8|p=221}}).
* If <math>K</math> is a [[Convex set|convex]] [[Balanced set|balanced]] [[Compact space|compact]] set in a locally convex space <math>Y</math>, then its [[Linear span|linear span]] <math>{\mathbb C}K=\operatorname{span}(K)</math> possesses a unique structure of a Smith space with <math>K</math> as the universal compact set (and with the same topology on <math>K</math>).{{sfn|Akbarov|2009|p=468}}
* If <math>M</math> is a (Hausdorff) compact topological space, and <math>{\mathcal C}(M)</math> the [[Banach space#Classical spaces|Banach space of continuous functions]] on <math>M</math> (with the usual sup-norm), then the stereotype dual space <math>{\mathcal C}^\star(M)</math> (of [[Radon measure]]s on <math>M</math> with the topology of uniform convergence on compact sets in <math>{\mathcal C}(M)</math>) is a Smith space. In the special case when <math>M=G</math> is endowed with a structure of a [[Topological group|topological group]] the space <math>{\mathcal C}^\star(G)</math> becomes a natural example of a stereotype group algebra.{{sfn|Akbarov|2003|p=272}}
* A [[Banach space]] <math>X</math> is a Smith space if and only if <math>X</math> is finite-dimensional.

==See also==
*[[Stereotype space]]
*[[Brauner space]]

== Notes ==
{{reflist}}

==References==

* {{cite journal|last=Smith|first=M.F.|title=The Pontrjagin duality theorem in linear spaces|journal=Annals of Mathematics|year=1952|volume=56|issue=2|pages=248–253|doi=10.2307/1969798|jstor=1969798}}
* {{cite journal|last=Akbarov|first=S.S.|title=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|s2cid=115297067|doi-access=free}}
* {{cite journal|last=Akbarov|first=S.S.|title=Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity|journal=Journal of Mathematical Sciences|year=2009|volume=162|issue=4|pages=459–586|arxiv=0806.3205|doi=10.1007/s10958-009-9646-1|s2cid=115153766}}
* {{cite thesis |type=PhD |last=Furber |first=R.W.J. |date=2017 |title=Categorical Duality in Probability and Quantum Foundations|publisher=Radboud University|url=http://people.cs.aau.dk/~furber/phdthesis.pdf}}

{{Functional analysis}}


[[Category:Functional analysis]]
[[Category:Topological vector spaces]]

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