Abstract
In this paper barrelled linearly topologized modules over an arbitrary discrete valuation ring are introduced. A general form of the Banach-Steinhaus theorem for continuous linear mappings on barrelled linearly topologized modules is established and some consequences of it are derived.
Highlights
Motivated by the fundamental works (Schwartz, 1950) (see (Schwartz, 1966)) and (Dieudonné & Schwartz, 1949), (Bourbaki, 1950) introduced the classical barrelled locally convex spaces, which may be regarded as the Hausdorff locally convex spaces characterized by the validity of the Banach-Steinhaus property, and proved certain important facts related to that class of spaces (a quite complete study of such spaces may be found in (Pérez Carreras & Bonet, 1987))
The main result established here is a general form of the Banach-Steinhaus theorem for continuous linear mappings on barrelled linearly topologized modules, from which the equicontinuity of separately equicontinuous sets of bilinear mappings between certain linearly topologized modules is derived
By using duality arguments, Hausdorff barrelled linearly topologized modules are characterized by the pertinent Banach-Steinhaus property
Summary
Mauro Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rio de Janeiro, Brasil 2 Centro Interdisciplinar de Ciências da Natureza, Universidade Federal da Integração Latino-Americana, Paraná, Brasil. Received: December 18, 2019 Accepted: January 13, 2020 Online Published: January 20, 2020 doi:10.5539/jmr.v12n1p69
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