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  "body": "\n## Definition\n\nLet $X$ be a real or complex [[Banach space|06GZ956657]].\nFor vectors $x_1, \\dots, x_n$ in $X$, write\n$$R(x_1, \\dots, x_n) = \\Big( \\frac{1}{2^n} \\sum_{\\varepsilon \\in \\set{-1, 1}^n} \\Big\\| \\sum_{i=1}^n \\varepsilon_i x_i \\Big\\|^2 \\Big)^{1/2},$$\nthe quadratic mean of the norm over all choices of signs.\n\n1. Let $1 \\le p \\le 2$. The space $X$ has *type* $p$ if there is a constant $C$ with $R(x_1, \\dots, x_n) \\le C \\big( \\sum_{i=1}^n \\norm{x_i}^p \\big)^{1/p}$ for all $n$ and all $x_1, \\dots, x_n \\in X$.\n2. Let $2 \\le q \u003c \\infty$. The space $X$ has *cotype* $q$ if there is a constant $C$ with $\\big( \\sum_{i=1}^n \\norm{x_i}^q \\big)^{1/q} \\le C \\, R(x_1, \\dots, x_n)$ for all $n$ and all $x_1, \\dots, x_n \\in X$.\n\n## Remarks\n\nExample: a Hilbert space $H$ has type $2$ and cotype $2$ with $C = 1$.\nExpanding the squares, the cross terms $\\pm 2 \\operatorname{Re} \\ip{x_i}{x_j}$ cancel in the average over signs.\nSo $R(x_1, \\dots, x_n)^2 = \\sum_i \\norm{x_i}^2$ exactly.\n\nNon-example: $\\ell^1$ has no type $p \u003e 1$.\nFor the unit vectors $e_1, \\dots, e_n$, every sign sum has norm $n$, so $R = n$.\nThe right side of the type inequality is $C n^{1/p}$, and $n \\le C n^{1/p}$ fails for large $n$.\n\nEvery Banach space has type $1$, by the triangle inequality.\nType $p$ implies type $p'$ for $p' \u003c p$, and cotype $q$ implies cotype $q'$ for $q' \u003e q$.\nBy the Kahane inequality, the quadratic mean in $R$ can be replaced by the plain mean of the norm, with other constants.\nFor $1 \\le p \u003c \\infty$, the space $L^p$ has type $\\min(p, 2)$ and cotype $\\max(p, 2)$.\n",
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