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Let $(a_k)_{k \ge 1} ⊂ \mathbb K$ be a bounded sequence and define $x_n = \frac{1}{n} \sum_{1 \leq k \leq n}a_k e_k$, where $(e_n)_{n \ge 1}$ is an orthonormal basis of a Hilbert Space H. Show that $\sqrt n x_n \to 0$ weakly for n → ∞.

I have proven another part of this exercise where I had to show that $x_n$ converges strongly to 0. But how can I show the weakly convergence of $\sqrt n x_n \to 0$?

user372904
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If $(\sqrt{n}x_n)_{n\in\mathbb N}$ is bounded, it suffices to show that $\langle \sqrt{n}x_n, e_k\rangle \to 0$ for each $k\in\mathbb N$ (see Boundedness and pointwise convergence imply weak convergence in $\ell^p$), which is easy to show.

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Lukas Betz
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