Given a Hilbert space H and a Hilbert basis $[{e_n}] _{n=1}^{\infty}$ in $H$ (which is a complete orthonormal system).
Prove there exists a vector $x$ in $H$ such that $<x, e_n > = \frac {1}{n}$ where $<\cdot,\cdot>$ is the inner product in $H$.
Is there a difference between a Hilbert space and an Inner Product space? What is the inner product in H? I seem to be confused since it is not explicitly defined. Any help is appreciated.
Note that, by Parseval's inequality, every element of $H$ is written uniquely as $x=\sum_n\langle x,e_n\rangle e_n$. Conversely, given any sequence $(\lambda_n)\in\ell^2$, the vector $x=\sum_n\lambda_ne_n$ is a well-defined element of $H$ and satisfies $\langle x,e_n\rangle=\lambda_n$.
What you are looking for follows directly by the above and the observation that the sequence $(\frac{1}{n})$ is a sequence in $\ell^2$, i.e. $\sum_n\frac{1}{n^2}<\infty$.
A Hilbert space is an inner product space that is complete under the induced norm $\|\cdot\|:=\sqrt{\langle\cdot,\cdot\rangle}$.