Fourier basis of L2 of the circle

Remarks

Orthonormality is a direct computation: $\langle e_n, e_m \rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i(n-m)x} \, dx$ is $1$ if $n = m$ and $0$ otherwise. Completeness means that the linear span of the $e_n$ is dense in $L^2(\mathbb{T})$. This span is the space of trigonometric polynomials; it is dense in $C(\mathbb{T})$ for the supremum norm by the density of trigonometric polynomials, and $C(\mathbb{T})$ is dense in $L^2(\mathbb{T})$. Since $\norm{g}_2 \le \sup \abs{g}$, uniform approximation gives approximation in $L^2$. The coordinates of $f$ in this basis are its Fourier coefficients $\hat f(n) = \langle f, e_n \rangle$. Example: $\cos x = \frac{1}{2} e_1 + \frac{1}{2} e_{-1}$, and indeed $\norm{\cos}_2^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} \cos^2 x \, dx = \frac{1}{2} = \frac{1}{4} + \frac{1}{4}$. With the normalised measure $dx / 2\pi$ the $e_n$ have norm $1$; with the measure $dx$ one uses $e_n / \sqrt{2\pi}$ instead.

Going deeper

For a student

The theorem puts Fourier series inside Hilbert space theory. General facts about orthonormal bases then give at once the convergence of Fourier series in $L^2$, Parseval's identity and the Riesz–Fischer theorem. The map $f \mapsto (\hat f(n))_{n \in \Z}$ is a unitary isomorphism from $L^2(\mathbb{T})$ onto $\ell^2(\Z)$. The same holds on the torus $\mathbb{T}^d$ with the monomials $e^{i k \cdot x}$, $k \in \Z^d$, and, by the Peter–Weyl theorem, on any compact abelian group with its characters.