Math 6420 • PDE

Let \((e_n)\) be a total orthonormal sequence in a Hilbert space \(H\). Let \((x_n)\subset H\)
be a sequence. Prove that the two following statements are equivalent.
\begin{enumerate}
\item[(i)] \(x_n\to x\) weakly.
\item[(ii)] The sequence \((x_n)\) is bounded and for any \(j\ge1\),
\(\langle x_n,e_j\rangle\to\langle x,e_j\rangle\) as \(n\to\infty\).
\end{enumerate}
\textbf{Hint:} Show that if (ii) holds and
\(y_N\in\operatorname{span}\{e_1,\ldots,e_N\}\) then
\(\langle x_n,y_N\rangle\to\langle x,y_N\rangle\) as \(n\to\infty\).