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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $V$ be a vector space over $\\mathbb{K}$.\nAn *inner product* on $V$ is a map $\\langle \\cdot, \\cdot \\rangle \\colon V \\times V \\to \\mathbb{K}$ such that, for all $x, y, z \\in V$ and $a, b \\in \\mathbb{K}$:\n\n1. $\\langle ax + by, z \\rangle = a \\langle x, z \\rangle + b \\langle y, z \\rangle$.\n2. $\\langle y, x \\rangle = \\overline{\\langle x, y \\rangle}$.\n3. $\\langle x, x \\rangle \u003e 0$ if $x \\ne 0$.\n\n## Remarks\n\nOver $\\R$, the conjugation does nothing, so an inner product is a symmetric positive definite bilinear form.\nOver $\\C$, conditions 1 and 2 give $\\langle x, by \\rangle = \\bar b \\langle x, y \\rangle$, so the map is conjugate linear in the second argument.\nMany physicists, and Mathlib, use the opposite convention: linear in the second argument.\nThe [[dot product|0S52TNNTH9]] on $\\R^n$ and $\\langle x, y \\rangle = \\sum_i x_i \\overline{y_i}$ on $\\C^n$ are inner products.\n\n## Explanations\n\n### L1 human\n\nAn inner product generalises the dot product to abstract vector spaces.\nIt lets us talk about lengths, angles and perpendicular vectors.\nOn the space of continuous functions on $[0, 1]$, the integral $\\int_0^1 f(t) g(t)\\, dt$ is an inner product.\n",
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