Let $E=\{e_{n}\}_{n\in\mathbb{N}}$ be an orthonormal basis in an infinite dimensional separable Hilbert space $H$. With help from here I managed to prove that $E\cup\{0_{H}\}$ is contained in the weak closure of $E$. Is the reverse inclusion also true?
My attempt to prove it: If $x$ is in the weak closure of $E$, then there is a net $(x_i)$ in $E$ that converges weakly to $x$. So $x_{i}$ is of the form $e_{n_i}$ for some $n_i\in\mathbb{N}$. Weak convergence implies that $\langle e_{n_i},h\rangle\to\langle x,h\rangle$ for all $h\in H$. How do I proceed? Any suggestions are greatly appreciated!
You know already that $\langle e_n,h\rangle\to 0$ as $n\to\infty$. So, as a subsequence, $\langle e_{n_i},h\rangle\to 0$ as $i\to\infty$ (or $n_i = k$ is constant for almost all $i$, in which case $e_{n_i}\to e_k$). So, $\langle x,h\rangle = 0$ for any $h\in H$, resulting in $x=0$.