Pontryagin duality in the theory of topological vector spaces (Q1907434)

Since the 1950s, Pontryagin-type duality theorems in the theory of topological vector spaces have been discovered, forgotten, and then rediscovered by different authors, for the most part working independently of each other. These results would now appear to have been consigned to oblivion, at least in Russia. Meanwhile, the class of Pontryagin-dual locally convex spaces possesses extremely remarkable properties that undoubtedly had been the reason why so many specialists had subjected this class to systematic study following the initial observations of M. F. Smith, B. S. Brudovskii, W. C. Waterhouse, and K. Brauner. In the present note we announce a number of results that extend some of the concepts put forward by the above authors. We will describe the properties of Pontryagin-dual spaces and provide material for subsequent studies where we will discuss certain applications of these results in the theory of homologies of topological algebras.