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Extended Schauder decompositions of locally convex spaces†

Published online by Cambridge University Press:  18 May 2009

J. H. Webb
J. H. Webb
Affiliation:
University of Cape Town, Rondebosch, C.P., South Africa
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Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 2 , September 1974 , pp. 166 - 171
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

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