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  "body": "\n## Statement\n\nLet $X$ be a normed space over $\\mathbb{R}$ or $\\mathbb{C}$.\nIf the closed unit ball $\\{ x \\in X : \\|x\\| \\le 1 \\}$ is [[compact|00R0X1WGPX]], then $X$ is finite-dimensional.\n\n## Remarks\n\nProof: if $X$ is infinite-dimensional, build unit vectors $x_1, x_2, \\ldots$ by [[Riesz's lemma|097C7KCAT5]], with $x_{n+1}$ at distance at least $\\tfrac{1}{2}$ from the span of $x_1, \\ldots, x_n$. That span is finite-dimensional, so it is [[complete|0ABP2F2ZP7]] and closed. Then $\\|x_m - x_n\\| \\ge \\tfrac{1}{2}$ for $m \\neq n$, so no subsequence converges, and the ball is not [[sequentially compact|0K8SGQSV7G]].\nWith the [[Heine–Borel theorem in finite dimension|0PHT6NBYPB]], we get: a normed space is finite-dimensional if and only if its closed unit ball is compact.\nEquivalently, a normed space is locally compact if and only if it is finite-dimensional.\n\n## Explanations\n\n### L1 human\n\nIn infinite dimensions, the closed unit ball is never compact: there is always room for infinitely many unit vectors that stay far apart.\nThis is one of the main differences between finite and infinite dimensions.\n",
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