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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional vector space over $\\mathbb{R}$ or $\\mathbb{C}$, with a norm $\\|\\cdot\\|$ and the metric $d(x, y) = \\|x - y\\|$.\nThen a subset $K \\subseteq V$ is [[compact|00R0X1WGPX]] if and only if $K$ is [[closed|0G5AV9Q4RR]] and [[bounded|0P953MG05E]].\n\n## Remarks\n\nProof: choose a basis. By [[equivalence of norms in finite dimension|03MC70ZRNR]], the norm is equivalent to the Euclidean norm of the coordinates. Equivalent norms have the same closed sets, bounded sets and compact sets. Then use the [[Heine–Borel theorem|07PFYV65MB]] in $\\mathbb{R}^n$ or $\\mathbb{C}^n = \\mathbb{R}^{2n}$.\nIn other words, each finite-dimensional normed space is a [[proper metric space|0Z4V98VN5E]].\nThe converse also holds: if the closed unit ball is compact, the space is [[finite-dimensional|0VZVV8VME5]].\n\n## Explanations\n\n### L1 human\n\nIn each finite-dimensional normed space, the compact sets are the closed bounded sets, just as in $\\mathbb{R}^n$.\n",
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        "url": "https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem"
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        "locator": "In the theory of metric spaces",
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