By Julian Lowell Coolidge

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There is a similar notion of bounded variation functions with values in a locally convex topological vector space. 98. Œ0; 1; V / . Proof. Clearly, bounded variation in the bornological sense is stronger than bounded variation in the topological sense. For a function f W Œ0; 1 ! f / W N ! in /, extended by 0 outside 42 1 Bornological vector spaces and inductive systems y V . By the range of I. N/ ˝ definition, f has bounded variation in the topological sense if and only if the set of y V . N/ and B Â V is a von Neumann bounded closed disk.

If V is complete, then any precompact subset of V is relatively compact. The closed disked hull of a relatively compact subset is compact and agrees with its complete disked hull. If f W V1 ! V2 is a bounded linear map, then f maps precompact, relatively compact, and compact subsets again to such subsets of V2 . 2]). Let V be a metrisable locally convex topological vector space. V /. Similar statements hold for compact and relatively compact subsets. 46. A bornology is called relatively compact or precompact if all bounded subsets are relatively compact or precompact, respectively.

B/ is finite for each n 2 N. x/ for all x 2 B. x/ x22 n E C2 n D D x uniformly for x 2 B. 54. Let V be a Fréchet space equipped with the precompact bornology. 53 applies, that is, a bounded linear map f W W ! 51. In contrast, if we equip V with the von Neumann bornology, then the notion of uniformly dense range for f W W ! V depends on the bornology on W . 51. If we equip CŒt with the fine bornology, then the map i W CŒt ! N/ instead, then i has uniformly dense range. Therefore, the property of having uniformly dense range depends on the bornology on W in this case.