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  "body": "\n## Example\n\nLet $n \\ge 1$ be an integer and let $p \\ge 1$ be a real number.\nFor $x = (x_1, \\ldots, x_n) \\in \\mathbb{R}^n$, let\n$$\\|x\\|_p = \\left( \\sum_{i=1}^{n} |x_i|^p \\right)^{1/p}, \\qquad \\|x\\|_\\infty = \\max_{1 \\le i \\le n} |x_i|.$$\nThen $\\|\\cdot\\|_p$ and $\\|\\cdot\\|_\\infty$ are norms on $\\mathbb{R}^n$.\nThey are $0$ only at $x = 0$, they satisfy $\\|\\lambda x\\| = |\\lambda| \\, \\|x\\|$ for real $\\lambda$, and they satisfy the triangle inequality $\\|x + y\\| \\le \\|x\\| + \\|y\\|$.\n\n## Remarks\n\nThe triangle inequality for $\\|\\cdot\\|_p$ is the Minkowski inequality.\nThe metric $\\|x - y\\|_p$ is the [[p-metric|0TFDZN6AXP]]. For $p = 2$, the norm is the Euclidean norm.\nFor each $x$, $\\|x\\|_p \\to \\|x\\|_\\infty$ as $p \\to \\infty$.\nAll these norms are [[equivalent|03MC70ZRNR]], and $\\mathbb{R}^n$ is a [[Banach space|0BDTTM24DP]] with each of them.\n\n## Explanations\n\n### L1 human\n\nThe p-norms give different ways to measure the length of a vector from its coordinates.\nFor $p = 1$, you add the absolute values. For $p = 2$, you get the usual length. For $p = \\infty$, you take the largest coordinate.\n",
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