Krein–Smulian theorem

In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan, who published them in 1940.{{cite journal|last1=Krein|first1=M.|author1-link=Mark Krein|last2=Šmulian|first2=V.|author2-link=Vitold Shmulyan|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 }}

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 $X$ be a Banach space and $K$ a weakly compact subset of $X$ (that is, $K$ is compact when $X$ is endowed with the weak topology). Then the closed convex hull of $K$ in $X$ is weakly compact.

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{{math theorem|name=Krein-Smulian Theorem{{sfn|Conway|1990|pp=159-165}}|note=|style=|math_statement=

Let $X$ be a Banach space and $A$ a convex subset of the continuous dual space $X^{\prime}$ of $X$ . If for all $r > 0,$ $A \cap \left\{x^{\prime} \in X^{\prime} : \left\| x^{\prime} \right\| \leq r\right\}$ is weak-* closed in $X^{\prime}$ then $A$ is weak-* closed.

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References

Bibliography

{{Conway A Course in Functional Analysis|edition=2}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Rudin Walter Functional Analysis|edition=2}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Trèves François Topological vector spaces, distributions and kernels}}

Krein–Smulian theorem

Statement

See also

References

Bibliography

Further reading