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  "body": "\n## Statement\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, with the [[induced norm|0MFJJV97RJ]] $\\norm{\\cdot}$.\nThen, for all $x, y \\in V$,\n$$\\norm{x + y}^2 + \\norm{x - y}^2 = 2 \\norm{x}^2 + 2 \\norm{y}^2.$$\n\n## Remarks\n\nThe proof expands both squares with the inner product; the mixed terms cancel.\nBy the Jordan–von Neumann theorem, a norm comes from an inner product if and only if it satisfies this law.\nThe norm $\\abs{x_1} + \\abs{x_2}$ on $\\R^2$ fails it for $x = (1, 0)$ and $y = (0, 1)$, so it comes from no inner product.\n\n## Explanations\n\n### L1 human\n\nIn a parallelogram, the squares of the two diagonals add up to the squares of the four sides.\nThe vectors $x + y$ and $x - y$ are the diagonals of the parallelogram with sides $x$ and $y$.\nIn a rectangle, both diagonals have the same length, and this is the Pythagorean theorem twice.\n",
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