Let $\{e_k\}_{k\in\Bbb N}$ be orthonormal basis of a inner product space over $\Bbb C$. I want to prove that $\langle x,y\rangle=\sum_{k=1}^\infty\langle x,e_k\rangle\overline{\langle y,e_k\rangle}$ for $x,~y\in\overline{\text{span}\{e_k\}_{k\in\Bbb N}}$.
My attempt: $x=\sum_{k=1}^\infty\langle x,e_k\rangle e_k$ and $y=\sum_{k=1}^\infty\langle y,e_k\rangle e_k$. Then $$\langle x,y\rangle=\left\langle\lim_{n\to\infty}\sum_{k=1}^n\langle x,e_k\rangle e_k,\lim_{n\to\infty}\sum_{k=1}^n\langle y,e_k\rangle e_k\right\rangle.$$ However, how to move out the $\lim$ symbol? I can't find a way to do this.
In order to move the limit out it suffices to show that the inner product is continuous. From the problem it looks like you're working in a Hilbert space or some related notion, not just an inner product space, over $\mathbb C,$ so you should have some kind of topology on your space already. This topology comes from the metric induced by the inner product, so the inner product is continuous almost by definition. Can you work out the details?