I have given a complex vector space $V$ together with a complex inner product $\langle\cdot,\cdot\rangle$.
Furthermore I have a sequence $(v_{n})_{n\in\mathbb{N}}\in V^{\mathbb{N}}$ which has the property that $\langle v_{n},w\rangle\to\langle v,w\rangle$ for all $w\in V$, where $v\in V$ and I also have a sequence $(c_{n})_{n\in\mathbb{N}}\in\mathbb{C}^{\mathbb{N}}$ which converges to $0$.
I have to show that $$\lim_{n\to\infty}\sum_{k=0}^{\infty}\vert c_{k}\vert^{2}\vert\langle x_{k},v_{n}-v\rangle\vert^{2}=0,$$
where $(x_{n})_{n\in\mathbb{N}}$ is a sequence of orthogonal elements in $V$, i.e. $\langle x_{n},x_{m}\rangle = \delta_{nm}$.
My Ansatz: If we are able to change the limit and the series, we are obviously done, since the condition $\langle v_{n},w\rangle\to\langle v,w\rangle$ for every $w$ exactly means that $\lim_{n\to\infty}\vert\langle x_{k},v_{n}-v\rangle\vert=0$.So my idea was to use dominated convergence, which means that we have to find some sequence $g_{n}$ such that
$$\forall n\in\mathbb{N}:\vert c_{k}\vert^{2}\vert\langle x_{k},v_{n}-v\rangle\vert^{2}\leq g_{k}$$ and $$\sum_{k=0}^{\infty}g_{k}<\infty.$$
(I hope this is right so far.)
For this reason I started to write
$$\vert c_{k}\vert^{2}\vert\langle x_{k},v_{n}-v\rangle\vert^{2}\leq \vert c_{k}\vert^{2}\Vert x_{k}\Vert^{2}(\Vert v_{n}\Vert^{2} - \Vert v\Vert^{2})=\vert c_{k}\vert^{2}(\Vert v_{n}\Vert^{2} - \Vert v\Vert^{2})$$
using Cauchy-Schwartz, the orthogonality of the $x_{n}$ as well as the triangle inequality. My next step would be that I know that every sequence with the property $\langle v_{n},w\rangle\to\langle v,w\rangle$ is bounded (a fact I know from the lecture) and hence:
$$\vert c_{k}\vert^{2}\vert\langle x_{k},v_{n}-v\rangle\vert^{2}\leq \vert c_{k}\vert^{2}(\sup_{n\in\mathbb{N}}\Vert v_{n}\Vert^{2} - \Vert v\Vert^{2})=\mathrm{const.}\vert c_{k}\vert^{2}$$
But the problem now is that I think that I have done something wrong, because I am not sure if one can say that
$$\sum_{k=0}^{\infty}\vert c_{k}\vert^{2}<\infty$$
only by knowing that $c_{k}\to 0$.
Any ideas?