Bornological linearly topologized modules over a discrete valuation ring

In this work the notion of a bornological linearly topologized module over a discrete valuation ring is introduced and it is shown that certain semimetrizable linearly topologized modules are bornological. The main result is a characterization of bornological linearly topologized modules, from which the completeness and the quasi-completeness of certain linearly topologized modules of continuous linear mappings are derived.