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Order and norm convergence in Banach lattices

Published online by Cambridge University Press:  18 May 2009

Andrew Wirth
Andrew Wirth
Affiliation:
Monash University, Clayton, Victoria, Australia, 3168
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Let(V, ≧, ‖ · ‖) be a Banach lattice, and denote V\{0} by V'. For the definition of a Banach lattice and other undefined terms used below, see Vulikh [4]. Leader [3] shows that, if norm convergence is equivalent to order convergence for sequences in V, then the norm is equivalent to an M-norm. By assuming the equivalence for nets in V we can strengthen this result.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 13
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloquium Publications 25, 3rd Edition (Providence, R. I., 1967), 366–377.Google Scholar
2.Kelley, J. L. and Namioka, I., Linear topological spaces (Princeton, 1963), 236–242.CrossRefGoogle Scholar
3.Leader, S., Sequential convergence in lattice groups, Problems in Analysis (ed. Gunning, R. C.), (Princeton, 1970), 273–290.Google Scholar
4.Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Groningen, 1967).Google Scholar