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On the Banach‐Steinhaus Theorem and the Continuity of Bilinear Mappings
Author(s) -
Beattie Ronald,
Butzmann H.P.
Publication year - 1991
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19911530126
Subject(s) - mathematics , locally convex topological vector space , banach space , bilinear interpolation , pure mathematics , eberlein–šmulian theorem , topological vector space , lp space , topological space , statistics
In this paper, the BANACH‐STEINHAUS theorem is extended from its usual locally convex topological vector space setting to the much broader framework of convergence vector spaces. It is used to derive theorems yielding the joint continuity of separately continuous bilinear mappings. These results are used, in turn, to show that the convolution mapping is a jointly continuous bilinear mapping when the distribution spaces ℰ and carry the canonical convergence vector space structures.