[proofplan] We define the Fourier coefficient map $\Phi: H \to \ell^2$, verify it is well-defined by [Bessel's Inequality](/theorems/540), isometric by Parseval's identity from the [Characterisation of Complete Orthonormal Systems](/theorems/541), and surjective by constructing the preimage of any $\ell^2$ sequence as a convergent series. Classification follows by composing isomorphisms through $\ell^2$. [/proofplan] [step:Define $\Phi$ and verify well-definedness, linearity, and the isometry property] Define $\Phi(x) = (c_1(x), c_2(x), \ldots)$ where $c_k(x) = (x, e_k)_H$. By [Bessel's Inequality](/theorems/540), $\sum c_k(x)^2 \leq \|x\|_H^2 < \infty$, so $\Phi(x) \in \ell^2$. Linearity follows from linearity of the inner product. Since $\{e_k\}$ is complete, Parseval's identity gives \begin{align*} \|\Phi(x)\|_{\ell^2}^2 = \sum_{k=1}^\infty c_k(x)^2 = \|x\|_H^2, \end{align*} so $\Phi$ is an isometry (hence injective). [/step] [step:Prove surjectivity by constructing the preimage of any $\ell^2$ sequence] [claim:Surjectivity Of Fourier Map] For every $(a_k) \in \ell^2$, the series $\sum a_k\, e_k$ converges in $H$ and $\Phi(\sum a_k e_k) = (a_k)$. [/claim] [proof] Define $S_n = \sum_{k=1}^n a_k\, e_k$. For $m > n$, orthonormality gives $\|S_m - S_n\|_H^2 = \sum_{k=n+1}^m a_k^2 \to 0$, so $\{S_n\}$ is Cauchy. By completeness of $H$, $S_n \to x$. For each $j$, $c_j(x) = (x, e_j)_H = \lim_n (S_n, e_j)_H = a_j$ (since $(S_n, e_j)_H = a_j$ for $n \geq j$). [/proof] [/step] [step:Classify all separable infinite-dimensional Hilbert spaces] If $H_1$ and $H_2$ are two separable infinite-dimensional Hilbert spaces, let $\Phi_1: H_1 \to \ell^2$ and $\Phi_2: H_2 \to \ell^2$ be isometric isomorphisms. Then $\Phi_2^{-1} \circ \Phi_1: H_1 \to H_2$ is an isometric isomorphism. [/step]