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  "body": "\n## Statement\n\nThere is a constant $K$ with the following property.\nLet $m, n \\ge 1$ and let $(a_{ij})$ be a real $m \\times n$ matrix.\nAssume that $\\abs{\\sum_{i,j} a_{ij} s_i t_j} \\le 1$ for all $s_1, \\dots, s_m, t_1, \\dots, t_n \\in \\set{-1, 1}$.\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] and let $u_1, \\dots, u_m, v_1, \\dots, v_n$ be vectors of norm at most $1$ in $H$.\nThen\n$$\\Big| \\sum_{i=1}^m \\sum_{j=1}^n a_{ij} \\ip{u_i}{v_j} \\Big| \\le K.$$\n\n## Remarks\n\nThe constant $K$ does not depend on $m$, $n$ or $H$.\nThe smallest such constant is the real Grothendieck constant $K_G$.\nIts exact value is unknown.\nKrivine proved $K_G \\le \\pi / (2 \\log(1 + \\sqrt{2})) \\approx 1.78$.\n\nOne proof uses random Gaussian vectors $g$.\nFor unit vectors $u$ and $v$, the average of $\\sgn \\ip{g}{u} \\sgn \\ip{g}{v}$ is $\\frac{2}{\\pi} \\arcsin \\ip{u}{v}$.\nSo signs drawn at random from the vectors reproduce a function of the [[inner products|0NB0AWGJBW]].\nKrivine first transforms the vectors so that this function becomes a fixed multiple of the inner product.\n\nThe inequality has a complex version, with a different constant.\nIn computer science it shows that a semidefinite relaxation approximates the cut norm of a matrix within a constant factor.\n\n## Explanations\n\n### L1 human\n\nReplace numbers $\\pm 1$ by unit vectors in some Euclidean space, and products by inner products.\nA sum built from a matrix can only grow by a bounded factor under this replacement, whatever the size of the matrix.\nThis links a hard combinatorial maximum over signs with an easy geometric maximum over vectors.\n\n### L2 human\n\nGrothendieck stated the result in his work on tensor products of Banach spaces.\nIn that language it says that every bounded linear map from $L^1$ to $L^2$ is absolutely summing.\nThe inequality is a key tool in the study of the [[projective tensor product|685ZV73JJ8]] and of operator ideals.\n",
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