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  "body": "\n## Summary\n\nAn LF-space is a [[locally convex space|B6GY2KJMWW]] obtained as the inductive limit of an increasing sequence of [[Fréchet spaces|3GAA614673]].\nThe letters stand for \"limit of Fréchet spaces\".\nThe main example is the space $C_c^\\infty(\\Omega)$ of test functions on an open set $\\Omega \\subseteq \\R^d$.\nIt is the union, over compact sets $K \\subseteq \\Omega$, of the Fréchet spaces of smooth functions supported in $K$.\nIts dual is the space of distributions.\nLF-spaces are [[barrelled|ZE6DXEJB8H]] and [[bornological|4R6SQD60FN]], although they are usually not metrizable.\n\n## Explanations\n\n### L1 human\n\nSome spaces are natural unions of a growing chain of nicer spaces.\nSmooth functions with compact support are the union of the smooth functions supported in $[-1, 1]$, in $[-2, 2]$, in $[-3, 3]$, and so on.\nEach piece is a Fréchet space with a good topology.\nThe LF topology glues these topologies together in the best convex way.\nFor test functions, a sequence converges when all its terms lie in one piece and it converges there.\nIn particular the supports must stay in a fixed compact set.\n",
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