Fourier Coefficients in arbitrary Hilbert Spaces

Question

Say we have an orthonormal basis $\{e_n\}$ for a infinite Hilbert Space $H$. I want to prove that any vector $x=\sum_{n=1}^\infty\langle x, e_n\rangle e_n$. I don't know where to start. Could I have any help?

Answer 1

What is the meaning of $$ x=\sum_{n=1}^\infty \langle x,e_n\rangle e_n? $$ This is $$ x=\lim_{m\to\infty}\sum_{n=1}^m \langle x,e_n\rangle e_n, $$ or in other words $$ \left\|x-\sum_{n=1}^m \langle x,e_n\rangle e_n\right\|\to 0, $$ as $m\to\infty$. Now recall $\|y\|^2=\langle y,y\rangle$, and try to show that the preceding limit goes to $0$.

Answer 2

Consider the set of all vectors of that form. Prove that it is a (closed) subspace, and that its orthogonal is zero. Then it has to be all of $H$.