The Eberlein-Smulian Theorem

# The Eberlein-Smulian Theorem

Theorem 1 (The Eberlein-Smulian Theorem): Let $X$ be a Banach space and let $B_X$ be the closed unit ball of $X$. Then $B_X$ is weakly compact if and only if $B_X$ is weakly sequentially compact. |

The Eberlein-Smulian theorem alongside Kakutani's Theorem gives us the following characterization of weak compactness for Banach spaces.

Theorem 2 (Characterization of Weak Compactness for Banach Spaces): Let $X$ be a Banach space and let $B_X$ be the closed unit ball of $X$. Then the following statements are equivalent:a) $X$ is reflexive.b) $B_X$ is weakly compact.c) $B_X$ is weakly sequentially compact. |

**Proof:**$(a) \Leftrightarrow (b)$ is Kakutani's theorem and $(b) \Leftrightarrow (c)$ is the Eberlein-Smulian theorem. $\blacksquare$