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  "body": "\n## Statement\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $(V, \\norm{\\cdot})$ be a [[normed vector space|03J5RAKB30]] over $\\mathbb{K}$.\nThen the following are equivalent.\n\n1. There is an inner product $\\ip{\\cdot}{\\cdot}$ on $V$ whose [[induced norm|0MFJJV97RJ]] is $\\norm{\\cdot}$, that is, $\\norm{x}^2 = \\ip{x}{x}$ for all $x \\in V$.\n2. The norm satisfies the parallelogram law: $\\norm{x + y}^2 + \\norm{x - y}^2 = 2 \\norm{x}^2 + 2 \\norm{y}^2$ for all $x, y \\in V$.\n\nWhen these hold, the inner product is unique and is given by the polarization identity.\nFor $\\mathbb{K} = \\R$ it is\n$$\\ip{x}{y} = \\frac{1}{4} \\left( \\norm{x + y}^2 - \\norm{x - y}^2 \\right),$$\nand for $\\mathbb{K} = \\C$, with $\\ip{\\cdot}{\\cdot}$ linear in the first argument, it is\n$$\\ip{x}{y} = \\frac{1}{4} \\sum_{k=0}^{3} i^k \\norm{x + i^k y}^2.$$\n\n## Remarks\n\nThe direction from 1 to 2 is the [[parallelogram law|026HPGYGS4]] in an [[inner product space|0QVNCJXRY3]].\nFor the converse in the real case, define $\\ip{x}{y}$ by the polarization formula.\nSymmetry and $\\ip{x}{x} = \\norm{x}^2$ are immediate.\nThe parallelogram law gives $\\ip{x + x'}{y} = \\ip{x}{y} + \\ip{x'}{y}$, so $t \\mapsto \\ip{tx}{y}$ is an [[additive function|KF719SZD5T]] of $t \\in \\R$.\nIt is continuous, because the norm is continuous, so it equals $t \\ip{x}{y}$.\nThe norm $\\abs{x_1} + \\abs{x_2}$ on $\\R^2$ fails the parallelogram law for $x = (1, 0)$, $y = (0, 1)$, since $4 + 4 \\ne 2 + 2$, so it comes from no inner product.\n\n## Explanations\n\n### L1 human\n\nSome ways of measuring length come from an inner product, which also measures angles, and some do not.\nThis theorem gives a simple test: a length comes from an inner product exactly when it obeys the parallelogram law.\nSo one identity about four sides and two diagonals decides whether the space has a full Euclidean geometry.\n\n### L2 human\n\nThe theorem shows that among the spaces $L^p([0, 1])$ with $1 \\le p \\le \\infty$, only $L^2([0, 1])$ has a norm that comes from an inner product.\nFor $p \\ne 2$ the parallelogram law fails for the indicator functions of $[0, 1/2]$ and $(1/2, 1]$.\nIt is a characterization of Hilbert spaces among Banach spaces, because completeness is not affected.\nThe proof is a functional equation argument: the parallelogram law is the [[quadratic functional equation|7MZGZFQ347]] for $f(x) = \\norm{x}^2$, and continuity turns additivity into linearity.\nThe law only involves two vectors at a time, so a norm is an inner product norm as soon as its restriction to every two-dimensional subspace is.\n",
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