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  "body": "\n## Statement\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, with the [[induced norm|0MFJJV97RJ]] $\\norm{\\cdot}$.\nIf $x, y \\in V$ are [[orthogonal|0Y2X7P2WB7]], then\n$$\\norm{x + y}^2 = \\norm{x}^2 + \\norm{y}^2.$$\n\n## Remarks\n\nThe proof: $\\norm{x + y}^2 = \\norm{x}^2 + 2 \\operatorname{Re} \\langle x, y \\rangle + \\norm{y}^2$.\nFor pairwise orthogonal $x_1, \\dots, x_k$, $\\norm{x_1 + \\cdots + x_k}^2 = \\norm{x_1}^2 + \\cdots + \\norm{x_k}^2$.\nIn a real space, the converse holds; in a complex space, the equality only gives $\\operatorname{Re} \\langle x, y \\rangle = 0$.\n\n## Explanations\n\n### L1 human\n\nThis is the theorem of Pythagoras for vectors.\nTwo orthogonal vectors are the legs of a right triangle, and their sum is the hypotenuse.\nIt holds in every inner product space, even in spaces of functions.\n",
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