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If $\{e_n| n \in \mathbb{N} \}$ is an orthonormal basis for a Hilbert space $\mathcal{H}$, then it would seem that $\langle x,\sqrt{n} e_n \rangle \to 0$ for all $x \in \mathcal{H}$, but the Banach-Steinhaus theorem implies that weakly convergent sequences have to be bounded. Could anyone give an example of an $x \in \mathcal{H}$ such that $\lim_{n \to \infty} \langle x,\sqrt{n} e_n \rangle \neq 0$ please?

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Let $x =\sum a_n e_n$ where $a_n=\frac 1 {\sqrt n}$ when $n=m^{2}$ for some $m$ and $0$ otherwise. Then $\langle x , \sqrt n e_n \rangle $ does not tend to $0$.

Note that $\sum a_n^{2} <\infty$ so $x$ is well defined.

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