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

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

See also

• Krein–Milman theorem – On when a space equals the closed convex hull of its extreme points
• Weak-* topology – Mathematical termPages displaying short descriptions of redirect targets