• Weyl's proof of the Bohr analogue of Parseval's identity for almost periodic functions. More precisely, let $f$ be a uniformly almost periodic $\mathbb C$-valued function on $\mathbb R$ and let $c_k=a(\lambda_k)$ be the corresponding sequence of the nontrivial Fourier constants of the function $f$ $$a(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t)e^{-i\lambda t}dt\neq 0$$ for some $\lambda=\lambda_k\in\mathbb R$ (it t}dt.$$It is known that the set \{\lambda\in \mathbb R:\ a(\lambda)\neq 0\} is at most countable)countable for any uniformly almost periodic function. Let c_k=a(\lambda_k)\neq 0 be the sequence of the nontrivial Fourier constants of the function f. Then$$\sum_{k}|c_k|^2=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|f(t)|^2dt.$$The proof is based on the spectral analysis of the operator$$Au=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t-s)u(s)dt.$$Weyl shows that A is a normal compact operator on the space of uniformly almost periodic functions (endowed with the scalar product (u,v)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}u(t-s)\overline{v(s)}dt). The result then follows from a clever application of the spectral theorem. A detailed exposition of the proof can be found in Theory of linear operators in Hilbert space by Akhiezer and Glazman (see Section 57). 2 added 1429 characters in body; deleted 2 characters in body One example is the • An operator-theoretic proof that Hamburger's the classical Hamburger moment problem admits a solution (see e.g. Methods of modern mathematical physics by Reed and Simon, volume 2, Theorem X.4). • Weyl's proof of the Bohr analogue of Parseval's identity for almost periodic functions. More precisely, let f be a uniformly almost periodic \mathbb C-valued function on \mathbb R and let c_k=a(\lambda_k) be the corresponding sequence of the nontrivial Fourier constants of the function f$$a(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t)e^{-i\lambda t}dt\neq 0 $$for some \lambda=\lambda_k\in\mathbb R (it is known that the set \{\lambda\in \mathbb R:\ a(\lambda)\neq 0\} is at most countable). Then$$\sum_{k}|c_k|^2=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|f(t)|^2dt.$$The proof is based on the spectral analysis of the operator$$Au=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t-s)u(s)dt. Weyl shows that $A$ is a normal compact operator on the space of uniformly almost periodic functions (endowed with the scalar product $(u,v)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}u(t-s)\overline{v(s)}dt$). The result then follows from a clever application of the spectral theorem.