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  "body": "\n## Statement\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $N$ be a bounded operator on $H$.\nAssume that $N$ is normal: $N N^* = N^* N$, where $N^*$ is the [[adjoint|6HSJBJ8TTQ]] of $N$.\nLet $\\sigma(N) \\subseteq \\C$ be the [[spectrum|BMKATJ802A]] of $N$.\nThen there is a unique [[projection-valued measure|ATGT25E5FF]] $E$ on the [[Borel sets|0WQXY32Q8D]] of $\\sigma(N)$ such that\n$$\\ip{N x}{y} = \\int_{\\sigma(N)} \\lambda \\, d E_{x,y}(\\lambda) \\quad \\text{for all } x, y \\in H,$$\nwhere $E_{x,y}(B) = \\ip{E(B) x}{y}$.\nIn short, $N = \\int_{\\sigma(N)} \\lambda \\, dE(\\lambda)$.\n\n## Remarks\n\nOne proof starts from the [[continuous functional calculus|JB7S6WPZ77]] $f \\mapsto f(N)$ on $C(\\sigma(N))$.\nFor fixed $x, y$, the map $f \\mapsto \\ip{f(N) x}{y}$ is a bounded linear functional on $C(\\sigma(N))$, so by the Riesz–Markov theorem it is integration against a complex measure $E_{x,y}$.\nOne then defines $E(B)$ by $\\ip{E(B) x}{y} = E_{x,y}(B)$ and checks that it is a projection-valued measure.\nThe projections $E(B)$ commute with every bounded operator that commutes with $N$.\nAn equivalent form holds when $H$ is separable.\nThen there are a $\\sigma$-finite measure space $(\\Omega, \\mu)$, a bounded measurable function $\\varphi$ on $\\Omega$ and a unitary map $U \\colon H \\to L^2(\\Omega, \\mu)$ with $U N U^{-1} f = \\varphi f$ for all $f \\in L^2(\\Omega, \\mu)$.\nExample: on $L^2([0,1])$, the operator $(Nf)(t) = t f(t)$ is self-adjoint with spectrum $[0,1]$, and $E(B)$ is multiplication by the indicator function of $B$.\nIn finite dimension, $E(B)$ is the orthogonal projection onto the sum of the eigenspaces for the eigenvalues in $B$, and the theorem is the [[spectral theorem for normal matrices|04CW1P44VD]].\nNormality is necessary: the right shift on $\\ell^2$ is not normal, and it is not of this form.\n\n## Explanations\n\n### L1 human\n\nA normal matrix is diagonal in a suitable orthonormal basis.\nAn operator on an infinite-dimensional space may have no eigenvectors, so a basis of eigenvectors may not exist.\nThe spectral theorem replaces the list of eigenvalues by a measure on the spectrum, whose values are projections.\nIn this sense, every normal operator is a continuous version of a diagonal matrix: it is multiplication by a function.\n\n### L2 human\n\nThe theorem makes it possible to define $f(N) = \\int f(\\lambda) \\, dE(\\lambda)$ for every bounded Borel function $f$ on $\\sigma(N)$, the Borel functional calculus.\nWith $f$ an indicator function, this gives the spectral projections of $N$, which split $H$ into pieces where $N$ acts with spectrum in a given set.\nThe self-adjoint case covers the observables of quantum mechanics with bounded values; the unitary case gives the spectral analysis of unitary operators, used in ergodic theory.\nFor unbounded self-adjoint operators, a similar theorem holds with a projection-valued measure on $\\R$.\n",
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