Problem 6

Exercise 8.6 (Closed subspaces are reflexive)   Show that a closed subspace of a reflexive Banach space is reflexive.

Proof. Let $S\subseteq X$ be a closed subspace of a reflexive space $X$. The theorem of Kakutani states that $S$ is reflexive if and only if the closed unit ball $B_S$ in $S$ is compact in the weak topology $\sigma(S,S^*)$. This topology is induced by the weak topology $\sigma(X,X^*)$. The theorem of Banach and Alaoglu states that $B_X$ is compact in the weak-* topology. Since $X$ is reflexive, the weak topology and weak-* topology coincide $\sigma(X,X^*)=\sigma(X^*,X)$. By compactness in the weak-* topology, $B_S\subseteq B_X$ is compact also in the weak topology because it is a weakly closed subset of a compact space. Therefore the theorem of Kakutani implies $S$ is reflexive. $\qedsymbol$