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  "body": "\n## Statement\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$, and let $L^2(\\mathbb{T})$ be the [[Lebesgue space|SASP14MA1Z]] of square integrable functions on $\\mathbb{T}$, with the inner product\n$$\\langle f, g \\rangle = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x) \\overline{g(x)} \\, dx.$$\nFor $n \\in \\Z$ let $e_n(x) = e^{inx}$ be the [[Fourier monomial|TZYSW0975T]] of index $n$.\nThen the family $(e_n)_{n \\in \\Z}$ is an [[orthonormal basis|0S9GGTR0EX]] of the [[Hilbert space|HM2HHE3J51]] $L^2(\\mathbb{T})$.\n\n## Remarks\n\nOrthonormality 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.\nCompleteness means that the linear span of the $e_n$ is dense in $L^2(\\mathbb{T})$.\nThis span is the space of trigonometric polynomials; it is dense in $C(\\mathbb{T})$ for the supremum norm by the [[density of trigonometric polynomials|KDRZKC9T1V]], and $C(\\mathbb{T})$ is dense in $L^2(\\mathbb{T})$.\nSince $\\norm{g}_2 \\le \\sup \\abs{g}$, uniform approximation gives approximation in $L^2$.\nThe coordinates of $f$ in this basis are its Fourier coefficients $\\hat f(n) = \\langle f, e_n \\rangle$.\nExample: $\\cos x = \\frac{1}{2} e_1 + \\frac{1}{2} e_{-1}$, and indeed $\\norm{\\cos}_2^2 = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\cos^2 x \\, dx = \\frac{1}{2} = \\frac{1}{4} + \\frac{1}{4}$.\nWith the normalised measure $dx / 2\\pi$ the $e_n$ have norm $1$; with the measure $dx$ one uses $e_n / \\sqrt{2\\pi}$ instead.\n\n## Explanations\n\n### L1 human\n\nThe pure waves $e^{inx}$ act like perpendicular unit axes in an infinite-dimensional space of functions.\nEvery square integrable periodic function has coordinates along these axes, namely its Fourier coefficients.\nCompleteness says that no direction is missing: a function perpendicular to every wave is zero.\n\n### L2 human\n\nThe theorem puts Fourier series inside Hilbert space theory.\nGeneral facts about orthonormal bases then give at once the convergence of Fourier series in $L^2$, Parseval's identity and the Riesz–Fischer theorem.\nThe map $f \\mapsto (\\hat f(n))_{n \\in \\Z}$ is a unitary isomorphism from $L^2(\\mathbb{T})$ onto $\\ell^2(\\Z)$.\nThe 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.\n",
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