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  "body": "\n## Definition\n\nLet $V$ be a vector space over $\\mathbb{R}$ or $\\mathbb{C}$.\nA norm on $V$ is a function $\\|\\cdot\\| \\colon V \\to [0, \\infty)$ such that, for all $x, y \\in V$ and all scalars $\\lambda$: $\\|x\\| = 0$ only for $x = 0$, $\\|\\lambda x\\| = |\\lambda| \\, \\|x\\|$, and $\\|x + y\\| \\le \\|x\\| + \\|y\\|$.\nTwo norms $\\|\\cdot\\|_a$ and $\\|\\cdot\\|_b$ on $V$ are equivalent when there are real numbers $c \u003e 0$ and $C \u003e 0$ with\n$$c \\, \\|x\\|_a \\le \\|x\\|_b \\le C \\, \\|x\\|_a \\quad \\text{for all } x \\in V.$$\n\n## Remarks\n\nEach norm gives a metric $d(x, y) = \\|x - y\\|$. Two norms are equivalent if and only if their metrics are strongly [[equivalent|0TWQ5PRKWS]].\nFor norms, equivalence is the same as giving the same [[metric topology|0A6YRNZB2R]].\nOn a finite-dimensional space, [[all norms are equivalent|03MC70ZRNR]].\nOn $C([0, 1])$, the sup norm and the norm $\\int_0^1 |f(t)| \\, dt$ are not equivalent.\n\n## Explanations\n\n### L1 human\n\nEquivalent norms measure the same vectors as small or large, up to a fixed factor.\nSo they agree on all questions of convergence and continuity.\n",
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