Let $H$ is a Hilbert space and $\left\{ {{e_j}} \right\}$ is an orthonormal basis. Put ${Z_k} = \overline {\bigoplus\limits_{j = k}^\infty {\left\langle {{e_j}} \right\rangle } } $ and $\left\{ {{u_k}} \right\}$ satisfies ${u_k} \in {Z_k}$ and $\left\| {{u_k}} \right\| = 1$ for all $k$. Prove that $u_k \to 0$ weakly in $H$.
My question arises when I study the Fountain theorem on Min-max Theorem book of M. Willem. I tried use the Riez theorem but I still did not obtain anythings.
We can write $$u_k=\sum\limits_{j=k}^{\infty} a_{jk} e_j$$ and $$1=\|u_k\|^{2} =\sum\limits_{j=k}^{\infty} |a_{jk}|^{2}.$$ For any $x=\sum x_ie_i$ in $H$ we have $$ \langle x, u_k \rangle$$ $$ =\sum\limits_{j=k}^{\infty} x_ja_{jk}$$ $$ \leq \sqrt { \sum\limits_{j=k}^{\infty} |x_j|^{2}} \sqrt { \sum\limits_{j=k}^{\infty} |a_{jk}|^{2}}$$ $$\leq \sqrt { \sum\limits_{j=k}^{\infty} |x_j|^{2}}\to 0$$ as $k \to \infty$ because $\sum |x_j|^{2} =\|x\|^{2} <\infty$.