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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional normed space and let $W$ be a normed space, both over $\\mathbb{R}$ or over $\\mathbb{C}$.\nThen each linear map $T \\colon V \\to W$ is [[continuous|0S3SMRH0Z5]].\n\n## Remarks\n\nProof: choose a basis $e_1, \\ldots, e_n$ of $V$. For $x = \\sum_i a_i e_i$, the triangle inequality gives $\\|Tx\\|_W \\le \\left( \\sum_i \\|T e_i\\|_W \\right) \\max_i |a_i|$. By [[equivalence of norms in finite dimension|03MC70ZRNR]], $\\max_i |a_i| \\le c \\|x\\|_V$ for some $c$. Thus $T$ is bounded, and so [[continuous|0F3K4C7XYW]].\nThe finite dimension of $W$ is not needed, only that of $V$.\nFor example, each matrix gives a continuous map $\\mathbb{R}^n \\to \\mathbb{R}^m$.\n\n## Explanations\n\n### L1 human\n\nIn finite dimensions, linear maps are automatically continuous, for each choice of norms.\nProblems with continuity only appear in infinite dimensions.\n",
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