Krein–Smulian theorem

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

Krein-Smulian Theorem:[2] 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.

Krein-Smulian Theorem:[2] Let X be a Banach space and A a convex subset of the continuous dual space ${\displaystyle X^{\prime }}$ of X. If for all r > 0, ${\displaystyle A\cap \left\{x^{\prime }\in X^{\prime }:\left\|x^{\prime }\right\|\leq r\right\}}$ is weak-* closed in ${\displaystyle X^{\prime }}$ then A is weak-* closed.

• Krein–Milman theorem
• Weak-* topology

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