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  "body": "\n## Statement\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, with the [[induced norm|0MFJJV97RJ]] $\\norm{\\cdot}$.\nLet $e_1, \\dots, e_k$ be an [[orthonormal family|0RW9TNNT87]] in $V$.\nThen, for each $x \\in V$,\n$$\\sum_{i=1}^{k} \\abs{\\langle x, e_i \\rangle}^2 \\le \\norm{x}^2.$$\n\n## Remarks\n\nThe proof applies the Pythagorean theorem to $x = \\sum_i \\langle x, e_i \\rangle e_i + r$, where $r$ is orthogonal to each $e_i$.\nEquality holds if and only if $x$ lies in the span of $e_1, \\dots, e_k$.\nThe inequality also holds for infinite orthonormal families, with a series on the left.\n\n## Explanations\n\n### L1 human\n\nThe squared lengths of the projections of a vector onto perpendicular directions add up to at most its squared length.\nThe missing part is the squared length of the component outside these directions.\nIf the directions span the whole space, then nothing is missing.\n",
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