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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $V$ be a vector space over $\\mathbb{K}$.\nA *norm* on $V$ is a map $\\norm{\\cdot} \\colon V \\to \\R$ such that, for all $x, y \\in V$ and $a \\in \\mathbb{K}$:\n\n1. $\\norm{x + y} \\le \\norm{x} + \\norm{y}$.\n2. $\\norm{ax} = \\abs{a} \\norm{x}$.\n3. If $\\norm{x} = 0$, then $x = 0$.\n\n## Remarks\n\nThese conditions imply $\\norm{x} \\ge 0$ for all $x$.\nOn $\\R^n$, the Euclidean norm $\\sqrt{x_1^2 + \\cdots + x_n^2}$, the norm $\\abs{x_1} + \\cdots + \\abs{x_n}$ and the norm $\\max_i \\abs{x_i}$ are norms.\nEach inner product gives a norm, see [[the induced norm|0MFJJV97RJ]].\nA norm defines a metric $d(x, y) = \\norm{x - y}$.\n\n## Explanations\n\n### L1 human\n\nA norm measures the length of a vector.\nThe length of a sum is at most the sum of the lengths, and scaling a vector scales its length.\nOnly the zero vector has length zero.\n",
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