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  "body": "\n## Definition\n\nLet $n \\ge 1$, and let $X$ and $Y$ be real [[normed vector spaces|03J5RAKB30]] of the same dimension $n$.\nThe *Banach–Mazur distance* between $X$ and $Y$ is\n$$d(X, Y) = \\inf_T \\norm{T} \\, \\norm{T^{-1}}.$$\nHere $T$ runs over the [[linear isomorphisms|0B8T10CE7N]] from $X$ to $Y$, and $\\norm{\\cdot}$ is the [[operator norm|06EKW1AKXF]].\n\n## Remarks\n\nAlways $d(X, Y) \\ge 1$, $d(X, Y) = d(Y, X)$ and $d(X, Z) \\le d(X, Y) \\, d(Y, Z)$; so $\\log d$ is a pseudometric.\nThe infimum is attained by compactness, and $d(X, Y) = 1$ if and only if $X$ and $Y$ are isometric.\nIn terms of the unit balls $B_X$ and $B_Y$, $d(X, Y)$ is the least $\\lambda \\ge 1$ with $B_X \\subseteq T B_Y \\subseteq \\lambda B_X$ for some linear isomorphism $T \\colon \\R^n \\to \\R^n$, after identifying both spaces with $\\R^n$.\nExample: let $X = \\ell_1^2$ and $Y = \\ell_\\infty^2$, that is, $\\R^2$ with the norms $\\abs{x_1} + \\abs{x_2}$ and $\\max(\\abs{x_1}, \\abs{x_2})$.\nThe map $T(x_1, x_2) = (x_1 + x_2, x_1 - x_2)$ satisfies $\\max(\\abs{x_1 + x_2}, \\abs{x_1 - x_2}) = \\abs{x_1} + \\abs{x_2}$, so it is an isometry and $d(X, Y) = 1$.\nNon-example: if $\\dim X \\ne \\dim Y$, there is no linear isomorphism, and the distance is not defined.\nJohn's theorem gives $d(X, \\ell_2^n) \\le \\sqrt{n}$ for every $n$-dimensional normed space $X$.\n",
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