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Weak and unbounded order convergence in Banach lattices

Published online by Cambridge University Press:  09 April 2009

A. W. Wickstead
A. W. Wickstead
Affiliation:
Department of Pure Mathematics, The Queen's University of Belfast.
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Abstract

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A net (xy) in a vector lattice is unbounded order convergent (uo-convergent) to 0 if u ∧ |xv| order converges to 0 for all u ≧ 0. We consider, in a Banach lattice, the relationship between weak and uo-convergence. We characterise those Banach lattices in which weak convergence implies uo-convergence and those in which uo-convergence of a bounded net implies weak convergence. Finally we combine the results to characterise those Banach lattices in which weak and uo-convergence coincide for bounded nets.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 24 , Issue 3 , November 1977 , pp. 312 - 319
Copyright
Copyright © Australian Mathematical Society 1977

References

DeMarr, R. (1964), ‘Partially ordered linear spaces and locally convex linear topological spaces’, Illinois J. Math. 8, 601–606.CrossRefGoogle Scholar
Köthe, G. (1969), Topological Vector Spaces I (Springer-Verlag, 1969).Google Scholar
Luxemburg, W. A. J. and Zaanen, A. C. (1964), ‘Notes on Banach function spaces X’, Indag. Math. 26, 493–506.CrossRefGoogle Scholar
Meyer-Nieberg, P. (1973), ‘Zur schwachen Kompaktheit in Banachverbänden’, Math. Z. 134, 303–315.Google Scholar
Schaefer, H. H. (1974), Banach Lattices and Positive Operators (Springer-Verlag, 1974).CrossRefGoogle Scholar
Walsh, B. (1968), ‘On characterising Köthe sequence spaces as vector lattices’, Math. Ann. 175, 253–256.CrossRefGoogle Scholar
Wickstead, A. W. (1975), ‘Compact subsets of partially ordered Banach spaces’, Math. Ann. 212, 271–284.CrossRefGoogle Scholar
Wirth, A. (1975), ‘Relatively uniform Banach lattices’, Proc. Amer. Math. Soc. 52, 178–180.CrossRefGoogle Scholar