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  "body": "\n## Summary\n\nLet $K$ be a [[totally bounded set|020Q06TYYH]] in a [[metric space|0VKD1TP7F3]].\nIts Kolmogorov $\\varepsilon$-entropy is the binary logarithm of the least number of pieces of diameter at most $2\\varepsilon$ needed to cover $K$.\nIt measures how many bits are needed to describe an element of $K$ to accuracy $\\varepsilon$.\nKolmogorov and Tikhomirov studied its growth as $\\varepsilon \\to 0$ for classes of smooth and analytic functions.\nIt gives lower bounds for every method of approximation or coding of the class, linear or not.\n\n## Explanations\n\n### L1 human\n\nSuppose you must store any element of a set $K$, but only up to an error $\\varepsilon$.\nCut $K$ into pieces so small that every piece has diameter at most $2\\varepsilon$, and store only the label of the piece.\nThe fewest pieces that work is the covering number, and its logarithm in base $2$ is the number of bits you need.\nFor example, the interval $[0, 1]$ needs $\\ceil{1/(2\\varepsilon)}$ pieces, so its $\\varepsilon$-entropy is about $\\log_2(1/\\varepsilon)$; a set of functions needs far more.\n",
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