After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l∞,τ(l∞,l1)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indEa, a∈A, ofseparable Banach spacesEa, and every Mackey null sequence inE is a null sequence in someEa. IfE is ultrabornological, thenE can be represented as lim indEa,a∈A, allEa separable Banach spaces, such that every fast null sequence inE is a null sequence in someEa.
Fast Convergence Convergence Condition Topological Vector Space Convex Space Separable Hilbert Space
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