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  "body": "\n## Statement\n\nLet $V$ be an [[inner product space|0QVNCJXRY3]] over $\\R$ or $\\C$, with the [[induced norm|0MFJJV97RJ]] $\\norm{\\cdot}$.\nThen, for all $x, y \\in V$,\n$$\\abs{\\langle x, y \\rangle} \\le \\norm{x} \\norm{y}.$$\nEquality holds if and only if $x$ and $y$ are linearly dependent.\n\n## Remarks\n\nFor the dot product on $\\R^n$, it says $\\left(\\sum_i x_i y_i\\right)^2 \\le \\left(\\sum_i x_i^2\\right)\\left(\\sum_i y_i^2\\right)$.\nIn a real inner product space, it allows the definition of the angle $\\theta \\in [0, \\pi]$ with $\\cos\\theta = \\langle x, y \\rangle / (\\norm{x} \\norm{y})$ for nonzero $x$ and $y$.\nOne proof uses that $\\norm{x - ty}^2 \\ge 0$ for all real $t$, so a quadratic has a nonpositive discriminant.\nIt implies the [[triangle inequality|0FJZMBX0ZZ]] for the induced norm.\n\n## Explanations\n\n### L1 human\n\nThe inner product of two vectors is at most the product of their lengths, in absolute value.\nIn the plane, this is the fact that $\\abs{\\cos\\theta} \\le 1$.\nEquality happens only when the vectors lie on one line.\n",
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