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  "body": "\n## Definition\n\nLet $V$ and $W$ be vector spaces over $\\mathbb{R}$ or $\\mathbb{C}$, with [[norms|0J3578562S]] $\\|\\cdot\\|_V$ and $\\|\\cdot\\|_W$.\nA map $T \\colon V \\to W$ is a continuous linear map when $T$ is linear and [[continuous|0SY9ECQ19Q]] for the metrics $\\|x - y\\|_V$ and $\\|x - y\\|_W$.\n\n## Remarks\n\nA linear map is continuous if and only if it is bounded, that is, $\\|Tx\\|_W \\le C \\|x\\|_V$ for some constant $C$ and all $x$. This is [[bounded linear maps are continuous|0F3K4C7XYW]].\nThe smallest such $C$ is the [[operator norm|0ZTEH055YF]] of $T$.\nOn a finite-dimensional space, [[each linear map is continuous|0C13E623WD]].\nThe derivative $p \\mapsto p'$ on polynomials, with the sup norm on $[0, 1]$, is linear but not continuous: $\\|x^n\\|_\\infty = 1$ but $\\|n x^{n-1}\\|_\\infty = n$.\n\n## Explanations\n\n### L1 human\n\nA continuous linear map respects the vector space structure and also the notion of closeness.\nIn infinite dimensions, a linear map can fail to be continuous, so this is a real condition.\n",
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