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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$, and let $(e_1, \\dots, e_n)$ be an [[orthonormal basis|0S9GGTR0EX]] of $V$.\nThen, for all $x, y \\in V$,\n$$\\langle x, y \\rangle = \\sum_{i=1}^{n} \\langle x, e_i \\rangle \\overline{\\langle y, e_i \\rangle} \\qquad \\text{and} \\qquad \\norm{x}^2 = \\sum_{i=1}^{n} \\abs{\\langle x, e_i \\rangle}^2.$$\n\n## Remarks\n\nIt is the case of equality in [[Bessel's inequality|0RBT5PB74K]].\nIn coordinates of an orthonormal basis, the inner product becomes the standard inner product of $\\mathbb{K}^n$.\nFor Fourier series, the same identity holds with infinite sums, for square integrable functions.\n\n## Explanations\n\n### L1 human\n\nIn an orthonormal basis, the inner product of two vectors is the standard inner product of their coordinates.\nIn particular, the squared length is the sum of the squared coordinates.\nThis is the Pythagorean theorem in $n$ dimensions.\n",
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