nLab Krein-Smulian theorem

Idea

The are two statements referred to as the Krein-Smulian theorem (also Krein-Šmulian or Krein-Smulyan):

The first says that with respect to the weak topology of Banach spaces $X$, the closed convex hull of a weakly compact subset $K \subset X$ is itself weakly compact.

The second says that…

References

Named after:

• Mark Krein, Vitold Šmulian: On Regularly Convex Sets in the Space Conjugate to a Banach Space, Annals of Mathematics 41 3 (1940) 556-583 [doi:10.2307/1968735, jstor:1968735]

Review:

• Robert Whitley: The Krein-Smulian Theorem, Proceedings of the AMS 97 2 (1986) 376-377 [doi:10.1090/S0002-9939-1986-0835903-X, pdf]

Textbook account:

• John B. Conway, section V.12 of: A Course in Functional Analysis, Graduate Texts in Mathematics 96, Springer (2007) [doi:10.1007/978-1-4757-4383-8]

See also:

• Wikipedia: Krein-Smulian theorem

• Antonio S. Granero: An extension of the Krein-Smulian theorem, Revista Matemática Iberoamericana 22 1 (2006) 93-110 [eudml:41966]