Bounded sets in (LF)-spaces

Bonet, José; Fernández, Carmen

doi:10.1090/S0002-9939-1995-1277098-2

Bounded sets in (LF)-spaces
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by José Bonet and Carmen Fernández

Proc. Amer. Math. Soc. 123 (1995), 3717-3723

DOI: https://doi.org/10.1090/S0002-9939-1995-1277098-2

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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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