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  "body": "\n## Statement\n\nLet $n, m \\ge 1$, let $S \\subseteq \\R^n$ be any subset, and let $L \\ge 0$.\nGive $\\R^n$ and $\\R^m$ the Euclidean norm.\nLet $f \\colon S \\to \\R^m$ satisfy the [[Lipschitz condition|0AW1E6YTJY]] $\\norm{f(x) - f(y)} \\le L \\norm{x - y}$ for all $x, y \\in S$.\nThen there is a map $F \\colon \\R^n \\to \\R^m$ with $F = f$ on $S$ and $\\norm{F(x) - F(y)} \\le L \\norm{x - y}$ for all $x, y \\in \\R^n$.\n\n## Remarks\n\nThe point is that the extension keeps the same Lipschitz constant $L$.\nFor $m = 1$ this follows from the [[McShane–Whitney extension|MV7ADC2DEH]].\nExtending each coordinate separately for $m \\ge 2$ only gives the constant $\\sqrt{m} \\, L$.\nThe theorem holds more generally for maps from a subset of one Hilbert space to another Hilbert space.\nIt can fail for norms that do not come from an [[inner product|0QVNCJXRY3]].\nOne proof first extends $f$ to one extra point, which reduces to a statement about intersections of balls in Euclidean space, and then uses Zorn's lemma or a compactness argument.\n\n## Explanations\n\n### L1 human\n\nA Lipschitz map does not stretch distances by more than a factor $L$.\nIf such a map is given only on part of space, one can fill in the rest without stretching any more.\nThis is easy for real-valued maps, but for vector-valued maps it needs the geometry of Euclidean space.\n\n### L2 human\n\nThe theorem is a basic tool in geometric measure theory, where Lipschitz maps are often defined on irregular sets.\nIt lets one assume that a Lipschitz map is defined on all of $\\R^n$, and then apply results such as [[Rademacher's theorem|QEY26RT19R]].\nThe key geometric step is about balls.\nTake two families of balls with the same radii, where the distances between centres in the second family are at most those in the first.\nIf the first family has a common point, then so does the second.\n",
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