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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional vector space over $\\mathbb{R}$ or $\\mathbb{C}$.\nThen any two norms on $V$ are [[equivalent|0J3578562S]].\n\n## Remarks\n\nProof idea: fix a basis and compare a norm $\\|\\cdot\\|$ with the [[sup norm|0KNJWEVSMG]] $\\|\\cdot\\|_\\infty$ of the coordinates. The triangle inequality gives $\\|x\\| \\le C \\|x\\|_\\infty$, so $\\|\\cdot\\|$ is continuous for $\\|\\cdot\\|_\\infty$. The unit sphere of $\\|\\cdot\\|_\\infty$ is compact by the [[Heine–Borel theorem|07PFYV65MB]], so $\\|\\cdot\\|$ has a positive minimum $c$ on it, by the [[extreme value theorem|0V4DET59NW]].\nThus there is only one norm topology on $\\mathbb{R}^n$, the Euclidean topology.\nThe statement fails in infinite dimensions: each infinite-dimensional normed space has norms that are not equivalent.\n\n## Explanations\n\n### L1 human\n\nIn finite dimensions, it does not matter which norm you choose: all norms give the same convergent sequences and the same open sets.\n",
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