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  "body": "\n## Summary\n\nA bornological space is a [[locally convex space|B6GY2KJMWW]] whose topology is determined by its [[bounded sets|3BAJ8ZTVPX]].\nIn such a space, every convex balanced set that absorbs all bounded sets is a neighbourhood of $0$.\nThe main consequence concerns a linear map from a bornological space to a locally convex space.\nSuch a map is continuous as soon as it maps bounded sets to bounded sets.\nEvery metrizable locally convex space, in particular every normed space, is bornological.\n\n## Explanations\n\n### L1 human\n\nFor linear maps between normed spaces, continuity is the same as sending bounded sets to bounded sets.\nBornological spaces are the locally convex spaces where this convenient test still works.\nThey are the spaces whose topology can be read off from the knowledge of which sets are bounded.\nA Banach space with its weak topology is a standard example that is not bornological when it is infinite-dimensional.\n",
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