Does the Fourier map preserve the inner product?

Question

Consider the Fourier map $\mathcal{F}: L^2(\mathbb{S}^1)\to \ell_2(\mathbb{Z})$. Here $L^2$ denotes $2\pi$-periodic functions on $\mathbb{R}$, and $(\mathcal{F} f)_n = (e_n, f)$, where $(-,-)$ is the inner product on $L^2$ and the $e_n = \frac{1}{\sqrt{2\pi}}e^{inx}$ are orthonormal to each other. It is known that this map is bijective, the proof uses (but is not immediate from) using the inverse map $\ell_2(\mathbb{Z})\to L^2$ that maps $\{c_n\}\mapsto\sum_{n=-\infty}^\infty c_ne_n$. Does this map preserve the inner product as well (and hence is an isomorphism of Hilbert spaces)?

Answer 1

Just so that this question can be closed, I comment that my question follows from Parseval's Theorem. [Credit to copper.hat and DisintegratingByParts].