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Countable enlargements of norm topologies and the quasidistinguished property

Published online by Cambridge University Press:  18 May 2009

F. X. Catalan  and
I. Tweddle
F. X. Catalan
Affiliation:
Department of MathematicsUniversity of StrathclydeLivingstone Tower26 Richmond StreetGlasgow G1 IXH.
I. Tweddle
Affiliation:
Department of MathematicsUniversity of StrathclydeLivingstone Tower26 Richmond StreetGlasgow G1 IXH.
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Let E be a Hausdorff locally convex space with continuous dual E1 and let M be a subspace of the algebraic dual E* such that M ∩ E1 = {0} and dim M = ℵ0. In the terminology of [4] the Mackey topology τ(E, E1 + M) is called a countable enlargement of τ(E, E1). There has been some interest in the question of when barrelledness is preserved under countable enlargements (see [4], [5], [6], [8], [9]). In this note we are concerned with the preservation of the quasidistinguished property for normed spaces under countable enlargements; this was posed as on open question by B. Tsirulnikov in [7]. According to [7] a Hausdorff locally convex space E is quasidistinguished if every bounded subset of its completion Ê is contained in the completion of a bounded subset of Ê (equivalently, in the closure in Ê of a bounded subset of E). Any normed space is clearly quasidistinguished and remains so under a finite enlargement (dim M < χ0) since the enlarged topology is normable. (See the Main Theorem of [7] for a general result on the preservation of the quasidistinguished property under finite enlargements.) We shall write QDCE for a countable enlargement which preserves the quasidistinguished property.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 2 , May 1993 , pp. 235 - 238
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

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