Let $V$ be an inner product space, and let $\left\{e_{n}\right\}_{n=1}^{\infty}$ be an orthonormal system.
Assume that $V$ is a complete inner product space (a Hilbert space). Does there exist $u \in V$ so that $\left\langle u, e_{n}\right\rangle=1 / \sqrt{n(n+2)} ?$
I am a little bit confused because of course I can say that $$u=\large{\sum_{n=1}^{n=\infty}}\frac{ e_n}{\sqrt{n(n+2)}}$$ But Is it fine that I am assuming that my vector is infity linear combination?
Yes, that is correct. Note that the series$$\sum_{n=1}^\infty\frac{e_n}{\sqrt{n(n+2)}}$$is a auchy series, and therefore it converges, since your space is a Hilbert space. And it is a Cauchy series since, if $m,n\in\Bbb N$ and $m\geqslant n$,$$\left\|\sum_{k=n}^m\frac{e_n}{\sqrt{k(k+2)}}\right\|^2=\sum_{k=n}^m\frac1{k(k+2)}$$and the series $\sum_{n=1}^\infty\frac1{n(n+2)}$ converges.