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  "body": "\n## Summary\n\nA Hilbert space is an [[inner product space|0QVNCJXRY3]] that is complete for the [[norm induced by its inner product|0MFJJV97RJ]].\nSo it is a [[Banach space|06GZ956657]] whose geometry has angles and orthogonality, as in Euclidean space.\nEvery Hilbert space has an orthonormal basis, and every closed subspace has an orthogonal complement.\nEvery continuous linear functional is the inner product with a fixed vector.\nHilbert spaces are the setting of Fourier series, of spectral theory and of quantum mechanics.\n\n## Explanations\n\n### L1 human\n\nA Hilbert space is a space of vectors, possibly of infinite dimension, where lengths and angles work as in the plane.\nThe theorem of Pythagoras and orthogonal projections still make sense.\nCompleteness means that a sequence whose terms get closer and closer always has a limit in the space.\nThe main example is $\\ell^2$, the sequences $(x_1, x_2, \\dots)$ with $\\sum \\abs{x_n}^2$ finite.\nAnother is the space $L^2$ of square-integrable functions, where Fourier series live.\n",
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