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  "body": "\n## Definition\n\nLet $V$ be an [[inner product space|0QVNCJXRY3]] over $\\R$ or $\\C$, with inner product $\\langle \\cdot, \\cdot \\rangle$.\nThe *norm induced by the inner product* is the map\n$$\\norm{x} = \\sqrt{\\langle x, x \\rangle} \\qquad (x \\in V).$$\n\n## Remarks\n\nIt is a [[norm|0V2WP9B1MF]]; the triangle inequality follows from the [[Cauchy–Schwarz inequality|0RFWW56Y22]].\nFor the dot product on $\\R^n$, it is the Euclidean length $\\sqrt{x_1^2 + \\cdots + x_n^2}$.\nIt satisfies the [[parallelogram law|026HPGYGS4]], and the [[polarisation identity|05FKHRT0B4]] recovers the inner product from it.\n\n## Explanations\n\n### L1 human\n\nThe length of a vector is the square root of its inner product with itself.\nIn the plane, this is the Pythagorean formula $\\sqrt{x^2 + y^2}$.\nSo every inner product gives a way to measure lengths.\n",
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