View source for Krein–Smulian theorem

In [[Mathematics|mathematics]], particularly in [[Functional analysis|functional analysis]], the Krein-Smulian theorem can refer to two theorems relating the closed [[Convex hull|convex hull]] and [[Compact space|compactness]] in the [[Weak topology|weak topology]]. They are named after [[Biography:Mark Krein|Mark Krein]] and [[Biography:Vitold Shmulyan|Vitold Shmulyan]], who published them in 1940.<ref>{{cite journal|last1=Krein|first1=M.|last2=Šmulian|first2=V.|doi=10.2307/1968735|journal=[[Annals of Mathematics]]|mr=2009|pages=556–583|series=Second Series|title=On regularly convex sets in the space conjugate to a Banach space|volume=41|year=1940|issue=3 |jstor=1968735 }}</ref>
==Statement==

Both of the following theorems are referred to as the Krein-Smulian Theorem.

{{math theorem|name=Krein-Smulian Theorem:{{sfn|Conway|1990|pp=159-165}}|note=|style=|math_statement=
Let <math>X</math> be a [[Banach space]] and <math>K</math> a weakly compact subset of <math>X</math> (that is, <math>K</math> is compact when <math>X</math> is endowed with the [[Weak topology|weak topology]]). Then the closed convex hull of <math>K</math> in <math>X</math> is weakly compact. 
}}

{{math theorem|name=Krein-Smulian Theorem{{sfn|Conway|1990|pp=159-165}}|note=|style=|math_statement=
Let <math>X</math> be a [[Banach space]] and <math>A</math> a convex subset of the continuous dual space <math>X^{\prime}</math> of <math>X</math>. If for all <math>r > 0,</math> <math>A \cap \left\{x^{\prime} \in X^{\prime} : \left\| x^{\prime} \right\| \leq r\right\}</math> is weak-* closed in <math>X^{\prime}</math> then <math>A</math> is weak-* closed. 
}}

==See also==

* {{annotated link|Krein–Milman theorem}}

==References==

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

* {{Conway A Course in Functional Analysis|edition=2}} <!--{{sfn|Conway|1990|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Rudin Walter Functional Analysis|edition=2}} <!--{{sfn|Rudin|1991|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->

{{Functional Analysis}}

{{DEFAULTSORT:Krein-Smulian theorem}}

[[Category:Banach spaces]]
[[Category:Normed spaces| ]]
[[Category:Topological vector spaces]]
[[Category:Theorems in functional analysis]]



{{Sourceattribution|Krein–Smulian theorem}}