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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $1 \\le p \\le \\infty$.\nThen the space [[$L^p(\\mu)$|SASP14MA1Z]] with the norm $\\norm{\\cdot}_p$ is complete.\nSo $L^p(\\mu)$ is a [[Banach space|06GZ956657]].\n\n## Remarks\n\nThe proof for $p \u003c \\infty$ starts from a Cauchy sequence $(f_n)$ and picks a subsequence with $\\norm{f_{n_{k+1}} - f_{n_k}}_p \u003c 2^{-k}$.\nThe function $g = \\abs{f_{n_1}} + \\sum_k \\abs{f_{n_{k+1}} - f_{n_k}}$ lies in $L^p$ by Minkowski's inequality and the [[monotone convergence theorem|G2S6H789RG]].\nSo the subsequence converges [[almost everywhere|RARTMA0ZEC]] to a function $f$, and [[Fatou's lemma|S95TR54TCG]] shows that $f \\in L^p$ and $f_n \\to f$ in $L^p$.\nA by-product: for $p \u003c \\infty$, every sequence that converges in $L^p$ has a subsequence that converges almost everywhere.\nThe case $p = \\infty$ is a direct argument with uniform convergence outside a null set.\n\nThe original form concerns $p = 2$.\nIf $(\\varphi_n)$ is an orthonormal sequence in $L^2(\\mu)$ and $(c_n)$ is a sequence of scalars with $\\sum_n \\abs{c_n}^2 \u003c \\infty$, then there is $f \\in L^2(\\mu)$ with $\\ip{f}{\\varphi_n} = c_n$ for all $n$.\nFor the trigonometric system on $[0, 2\\pi]$, every square-summable sequence is the sequence of Fourier coefficients of some square-integrable function.\n\nThe Lebesgue integral is essential here.\nThe continuous functions on $[0, 1]$ with the norm $\\int_0^1 \\abs{f}\\, dx$ do not form a complete space.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the spaces $L^p$ have no holes.\nIf the functions of a sequence get closer and closer to each other in the $L^p$ distance, then they get close to an actual function of the space.\nThis is the same property that the real numbers have and the rational numbers lack.\nIt is the main reason why analysts use the Lebesgue integral and not the Riemann integral.\n\n### L2 human\n\nCompleteness makes the tools of Banach space theory available on $L^p$: the Baire category theorem, the open mapping theorem and the uniform boundedness principle.\nFor $p = 2$ it makes $L^2(\\mu)$ a Hilbert space, which is the setting of Fourier series and of quantum mechanics.\nThe proof also gives the useful fact that $L^p$ convergence implies almost everywhere convergence along a subsequence, but not for the whole sequence.\nThe \"typewriter\" sequence of indicator functions of shrinking intervals that sweep across $[0, 1]$ converges to $0$ in $L^1$ but at no point.\n",
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  "name": "Riesz–Fischer theorem",
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