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.


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