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  "body": "\n## Statement\n\nLet $V$ be a vector space over $\\R$ or $\\C$ with a [[norm|0V2WP9B1MF]] $\\norm{\\cdot}$.\nThen, for all $x, y \\in V$,\n$$\\norm{x + y} \\le \\norm{x} + \\norm{y} \\qquad \\text{and} \\qquad \\bigl| \\norm{x} - \\norm{y} \\bigr| \\le \\norm{x - y}.$$\n\n## Remarks\n\nThe first inequality is an axiom of a norm; the second, the *reverse triangle inequality*, follows from it.\nFor the norm induced by an inner product, the first inequality follows from the [[Cauchy–Schwarz inequality|0RFWW56Y22]].\nIn $\\R^n$, it says that one side of a triangle is at most the sum of the other two sides.\n\n## Explanations\n\n### L1 human\n\nThe straight way from $0$ to $x + y$ is never longer than the way through $x$.\nThis is the triangle inequality.\nThe reverse form says that two vectors with a small difference have almost equal lengths.\n",
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