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.[1]

Statement

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

Krein-Smulian Theorem:[2]  Let be a Banach space and a weakly compact subset of (that is, is compact when is endowed with the weak topology). Then the closed convex hull of in is weakly compact.

Krein-Smulian Theorem[2]  Let be a Banach space and a convex subset of the continuous dual space of . If for all is weak-* closed in then is weak-* closed.

See also

  • Krein–Milman theorem – On when a space equals the closed convex hull of its extreme points
  • Weak-* topology – Mathematical term

References

    1. Krein, M.; Šmulian, V. (1940). "On regularly convex sets in the space conjugate to a Banach space". Annals of Mathematics. Second Series. 41 (3): 556–583. doi:10.2307/1968735. JSTOR 1968735. MR 0002009.
    2. Conway 1990, pp. 159–165.

    Bibliography


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