Can weak convergence be checked on an orthonormal basis?

Question

Let $H$ be a Hilbert space with orthnormal basis $(e_i)_{i\in I}$. Let $(x_n)_{n\in\mathbb N}$ be a sequence in $H$ with $\langle x_n, e_i\rangle \to \langle x, e_i\rangle$ for some $x\in H$ and all $i\in I$. Can we conclude from this that that $(x_n)_{n}$ converges weakly?

My intuition says no but I didn't find a counterexample.

Answer 1

Yes, if you know that your sequence (or net) is bounded. See here and here for more general results. For unbounded nets you may still have pointwise convergence to 0, without being weakly convergent, so in general the answer is no.