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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a [[locally convex space|B6GY2KJMWW]] over $\\mathbb{K}$.\nA subset $A \\subseteq E$ *absorbs* a subset $B \\subseteq E$ if there is $r \u003e 0$ such that $B \\subseteq cA$ for every $c \\in \\mathbb{K}$ with $\\abs{c} \\ge r$.\n$E$ is *bornological* if every [[convex|02AS45T3H9]] [[balanced|2TDSHJJYS4]] subset of $E$ that absorbs every [[bounded set|3BAJ8ZTVPX]] of $E$ is a [[neighbourhood|0ZCY0Z80B6]] of $0$.\n\n## Remarks\n\nEquivalently, $E$ is bornological if and only if every linear map from $E$ to a locally convex space that maps bounded sets to bounded sets is continuous.\nExample: every normed space $E$ is bornological.\nIf a convex balanced set $A$ absorbs the closed unit ball $B$, then $B \\subseteq c A$ for some $c \u003e 0$, so $A$ contains the ball of radius $1 / c$.\nMore generally, every metrizable locally convex space is bornological.\nNon-example: let $E$ be an infinite-dimensional Banach space with its weak topology $\\sigma(E, E')$.\nIts weakly bounded sets are norm bounded, by the uniform boundedness principle, so the closed unit ball absorbs all of them.\nBut the ball is not a weak neighbourhood of $0$, since every weak neighbourhood of $0$ contains a nonzero linear subspace.\n",
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