# Determining $x$ in a Hilbert space s.t $\langle x,e_n \rangle = 1/\sqrt{n}$, where $(e_n)_{n=1}^{\infty}$ is an orthonormal basis.

I have a quick question regarding Hilbert spaces:

Given a Hilbert space, $H$, with an orthonormal basis, $(e_n)_{n=1}^{\infty}$, is it possible to determine $x \in H$ s.t $\langle x,e_n\rangle=\dfrac{1}{\sqrt{n}}$ for all $n \in \mathbb{N}$?

How would one go about determining whether or not this is true? My intuition says that it is not possible, but I don't really know how to show it.

Any help and tips would be much appreciated, as I'm really stuck.

Thanks :-)

No, it is not. The only chance would be $$x = \sum_{n\geq 1}\frac{e_n}{\sqrt{n}}$$ but in such a case: $$\langle x,x\rangle = \sum_{n\geq 1}\frac{1}{n}$$ where the RHS is a divergent series.