Proof that weak convergence implies weak* convergence.

February 09, 2022 / Adam Frank

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.

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categories / real analysis theorem
tags / convergence, weak convergence, weak* convergence, dual space, bounded linear functionals, normed linear spaces, functional analysis

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Proof that weak convergence implies weak* convergence.