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  "body": "\n## Summary\n\nType and cotype measure how far a [[Banach space|06GZ956657]] is from a Hilbert space.\nThey compare random sign sums $\\sum_i \\pm x_i$ with the sizes of the $x_i$.\nA space of type $p$ makes such sums small, and a space of cotype $q$ makes them large.\nA theorem of Kwapień says that a Banach space has type $2$ and cotype $2$ exactly when it is isomorphic to a [[Hilbert space|HM2HHE3J51]].\nThese notions organise much of the local theory of Banach spaces, around results such as [[Dvoretzky's theorem|NRPQP49N9F]].\n\n## Explanations\n\n### L1 human\n\nTake some vectors and add them with random signs.\nIn the plane with the usual length, the average squared length of the result is the sum of the squared lengths, by the theorem of Pythagoras.\nIn other spaces the random sum can be longer or shorter than this.\nType and cotype record the best inequalities between the two sides.\nFor example, in $\\ell^1$ the unit vectors $e_1, \\dots, e_n$ give a random sum of norm exactly $n$, much larger than the Pythagorean value $\\sqrt{n}$.\n",
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