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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bounded sets in (LF)-spaces


Authors: José Bonet and Carmen Fernández
Journal: Proc. Amer. Math. Soc. 123 (1995), 3717-3723
MSC: Primary 46A13
MathSciNet review: 1277098
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Abstract: The behaviour of bounded sets is important in the theory of countable inductive limits of Fréchet spaces, the (LF)-spaces, and its applications. An (LF)-space is called regular if every bounded set is contained and bounded in one of the steps. In the present paper necessary conditions and sufficient conditions are given for the regularity of an (LF)-space. The conditions are expressed in terms of the behaviour of the neighbourhoods of the steps. It is proved that the conditions are equivalent for (LF)-spaces of sequences or of continuous functions.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1277098-2
PII: S 0002-9939(1995)1277098-2
Keywords: Fréchet spaces, (LF)-spaces, regularity and completeness, condition (M) of Retakh, Köthe sequence (LF)-spaces, weighted inductive limits
Article copyright: © Copyright 1995 American Mathematical Society