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  "body": "\n## Summary\n\nSpectral theory studies linear operators through their spectra, the sets of numbers $\\lambda$ for which $T - \\lambda I$ fails to be invertible.\nIt extends the theory of eigenvalues and diagonalisation of matrices to operators on infinite-dimensional spaces.\nIts central results, the spectral theorems, describe self-adjoint and normal operators as multiplication operators, and they are the mathematical basis of quantum mechanics.\n\n## Explanations\n\n### L1 human\n\nA symmetric matrix can be described completely by its eigenvalues and eigenvectors: in the right coordinates, it just stretches each axis by a number.\nSpectral theory asks how much of this survives for operators on spaces of functions or sequences.\nFor example, the operator that multiplies a square-integrable function $f(x)$ on $[0, 1]$ by $x$ has no eigenvectors at all, yet its spectrum is the whole interval $[0, 1]$.\nSpectral theory gives the language to say in what sense this operator still \"stretches by $x$ at the point $x$\".\n",
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