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On the completion of certain topological modules. (English) Zbl 0761.46041
The main result of this paper is an extension, in the case of topological modules, of the classical Grothendieck theorem on the completion of the dual of a locally convex space and similar results obtained by various authors in the case of non-locally convex topological vector spaces. Let $$A$$ be a separated topological ring, $$E$$ and $$F$$ two $$A$$-modules, $$H$$ a submodule of the $$A$$-module $$L(E,F)$$ of all $$A$$-linear mappings from $$E$$ to $$F$$. Supose $$F$$ a separated topological $$A$$-module and $${\mathcal V}_ F(0)$$ a base of neighborhoods of 0 in $$F$$. Let $${\mathcal B}$$ be a bornology on $$E$$ and $$\tau_ B$$ the topology of $${\mathcal B}$$-convergence on $$H$$. The author gives a precise description of the completion of $$(H,\tau_ B)$$ using elements of $${\mathcal B}$$ and $${\mathcal V}_ F(0)$$.

##### MSC:
 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)