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  "body": "\n## Statement\n\nLet $X$ be a [[metric space|0VKD1TP7F3]], let $A \\subseteq X$ be a [[closed|0ZCK8DQYN1]] subset, and let $L$ be a [[locally convex|B6GY2KJMWW]] topological vector space over $\\R$.\nLet $f \\colon A \\to L$ be a [[continuous|0VGFHTM75Y]] map.\nThen there is a continuous map $F \\colon X \\to L$ with $F(a) = f(a)$ for all $a \\in A$.\nMoreover, $F$ can be chosen with $F(X)$ contained in the [[convex hull|0FQSDC2EQ2]] of $f(A)$.\n\n## Remarks\n\nThe proof covers $X \\setminus A$ by small open balls whose radii are comparable to the distance to $A$.\nIt then takes a locally finite open refinement, which exists because metric spaces are paracompact.\nFor each set $W$ of the refinement, pick a point $a_W \\in A$ close to $W$, and let $(\\varphi_W)$ be a partition of unity subordinate to the refinement.\nSet $F(x) = \\sum_W \\varphi_W(x) f(a_W)$ for $x \\notin A$ and $F = f$ on $A$.\nContinuity at points of $A$ uses that $a_W$ is near $x$ when $W$ is near $x$, together with the local convexity of $L$.\n\nExample: with $L = \\R$, the theorem gives the [[Tietze extension theorem|0X52TNMTCJ]] for metric spaces, with the bound $\\inf f \\le F \\le \\sup f$.\n\nThe hypothesis that $A$ is closed is necessary.\nOn $X = [0, 1]$, the map $f(x) = \\sin(1/x)$ is continuous on $A = (0, 1]$, but it has no continuous extension to $X$, because $f$ has no limit at $0$.\n\n## Explanations\n\n### L1 human\n\nIf a continuous function is known on a closed part of a metric space, one can always fill in its values on the rest of the space continuously.\nThis holds not only for real values, but for values in any space where one can form averages in a controlled way.\nThe filled-in values are averages of known values, so they stay inside the convex hull of the original values.\n\n### L2 human\n\nThe theorem extends the Tietze extension theorem from real-valued to vector-valued maps, at the price of assuming that $X$ is metric.\nIts main consequence is that every convex subset of a locally convex metrizable topological vector space is an [[absolute retract|DK3MT3HYX5]].\nThe proof is a model use of partitions of unity on paracompact spaces, and the same technique appears in the [[Michael selection theorem|3RJ1K36A9A]].\n",
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