Corollaries to Kakutani's Theorem

# Corollaries to Kakutani's Theorem

Recall from the Kakutani's Theorem page that Kakutani's theorem states that if \$X\$ is a Banach space then \$X\$ is reflexive if and only if the closed unit ball \$B_X\$ of \$X\$ is weakly compact.

We will now look at some important corollaries to Kakutani's theorem.

 Corollary 1: Let \$X\$ be a Banach space. If \$X\$ is reflexive then every norm closed, bounded, and convex subset \$K\$ of \$X\$ is weakly compact.
• Proof: By Kakutani's theorem we have that the closed unit ball \$B_X\$ of \$X\$ is weakly compact. Therefore any norm closed ball of \$X\$ is weakly compact.
• Let \$K \subseteq X\$ be norm closed, bounded, and convex. By Mazur's Theorem, since \$K\$ is convex and norm closed, it is weakly closed. Since \$K\$ is bounded, there exists a norm closed ball \$B\$ such that \$K \subseteq B\$.
• We have established that \$B\$ must be weakly compact. Therefore \$K\$ is a weakly closed subset of a weakly compact set \$B\$ and thus \$K\$ is weakly compact. \$\blacksquare\$
 Corollary 2: Let \$X\$ be a Banach space. If \$X\$ is reflexive then the closed unit ball \$B_{X^{*}}\$ of \$X^*\$ is weak-* sequentially compact.
• Proof: Since \$X\$ is reflexive we have that the weak topology on \$X^*\$ is the same as the weak-* topology on \$X^*\$.
• By Alaoglu's Theorem we have that the closed unit ball \$B_{X^{*}}\$ of \$X^*\$ is weak-* compact, and so it is weakly compact.
• Since \$X^*\$ is a Banach space and \$B_{X^{*}}\$ is weakly compact, we have by Kakutani's theorem that \$X^*\$ is reflexive.
• From the theorem on the Every Bounded Sequence in a Reflexive Space X has a Weakly Convergent Subsequence page, we have that every bounded sequence in \$X^*\$ has a weak-* convergent subsequence. Since \$B_{X^*}\$ is bounded, we have that every sequence in \$B_{X^*}\$ has a weak-* convergent subsequence. Moreover, since \$B_{X^{*}}\$ is weak-* closed, we have that every sequence in \$B_{X^*}\$ has a weak-* convergent subsequence that converges IN \$B_{X^*}\$.
• Hence \$B_{X^*}\$ is weak-* sequentially compact. \$\blacksquare\$