Proof that weak convergence implies weak* convergence.

Theorem: Let \((X,\|\cdot\|)\) be a normed linear space and \(\{\varphi_n\}\cup\varphi\subseteq X^*\) and \(\{\varphi_n\}\overset w \to \varphi\). Then \(\varphi_n\overset *\to \varphi\).

Proof: Let \(f\in X\) and we show that \(\varphi_n(f)\to\varphi(f)\). Notice that by the assumption of weak convergence, if \(\psi\in X^{**}\) then \(\psi(\varphi_n)\to\psi(\varphi)\). We may therefore set \(\psi_f(\varphi')=\varphi'(f)\) in which case it follows that $$ \varphi_n(f) = \psi_f(\varphi_n)\to \psi_f(\varphi)=\varphi(f) $$ This shows \(\varphi_n\overset*\to\varphi\) as desired.