Math 6420 • PDE

Let \(X\) be a Banach space. Let \((f_n)\) be a sequence in \(X'\).
\begin{enumerate}
\item[(a)] Show that \(f_n\to f\) weakly in \(X'\) implies that for all \(x\in X\), we have
\(f_n(x)\to f(x)\).

\item[(b)] If there is an \(f\in X'\) such that \(f_n(x)\to f(x)\) for all \(x\in X\), does
\(f_n\to f\) weakly in \(X'\)? Explain your answer.
\end{enumerate}