The Banach-Steinhaus theorem

R. Beattie and H.-P. Butzmann
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R. Beattie: Mount Allison University, Department of Mathematics and Computer Science
H.-P. Butzmann: Universität Mannheim, Fakultät für Mathematik und Informatik

Chapter Chapter 7 in Convergence Structures and Applications to Functional Analysis, 2002, pp 195-206 from Springer

Abstract: Abstract While the content of the classical Banach-Steinhaus theorem varies somewhat in the literature, one very common variation is the following: if E and F are locally convex topological vector spaces and E is barrelled, then every pointwise bounded subset of 𝓛(E, F) is equicontinuous. This powerful theorem is used, for example to show that the pointwise limit of a sequence of continuous linear mappings is a continuous linear mapping. It is used as well to derive the continuity of separately continuous bilinear mappings.

Keywords: Vector Space; Topological Vector Space; Bilinear Mapping; Inductive Limit; Bounded Subset (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-94-015-9942-9_7

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