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Bornological versus topological analysis in metrizable spaces

Given a metrizable topological vector space, we can also use its von Neumann bornology or its bornology of precompact subsets to do analysis. We show that the bornological and topological approaches are equivalent for many problems. For instance, they yield the same concepts of convergence for sequences of points or linear operators, of continuity of functions, of completeness and completion. We also show that the bornological and topological versions of Grothendieck's approximation property are equivalent for Frechet spaces. These results are important for applications in noncommutative geometry. Finally, we investigate the class of ``smooth'' subalgebras appropriate for local cyclic homology and apply some of our results in this context.