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On weak approximation and convexification in weighted spaces of vector-valued continuous functions

Published online by Cambridge University Press:  18 May 2009

Marek Nawrocki
Marek Nawrocki
Affiliation:
Institute of Mathematics, A. Mickiewicz University, Matejki 48/49, 60–769 Poznan, Poland
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Let X be a completely regular Hausdorff space. A Nachbin family of weights is a set V of upper-semicontinuous positive functions on X such that if u, υ ∈ V then there exists w ∈ V and t > 0 so that u, υ ≤ tw. For any Hausdorff topological vector space E, the weighted space CV0(X, E) is the space of all E-valued continuous functions f on X such that υf vanishes at infinity for all υ ∈ V. CV0(X, E) is equipped with the weighted topologywv = wv(X, E) which has as a base of neighbourhoods of zero the family of all sets of the form

where υ ∈ Vand W is a neighbourhood of zero in E. If E is the scalar field, then the space CV0(X, E) is denoted by CV0(X). The reader is referred to [4, 6, 8] for information on weighted spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 1 , January 1989 , pp. 59 - 64
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

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