  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.
