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On order continuous norms

Author(s): Si-Kit Chung; Denny Leung
Journal: Proc. Amer. Math. Soc. 128 (2000), 1971-1974.
MSC (1991): Primary 46E30
Posted: November 24, 1999
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Abstract: It is shown that a normed vector lattice $(E,\|\cdot\|)$ is order continuous if and only if, for every lattice norm $\rho$ on $E$ with $\rho\leq \|\cdot\|$, the $\|\cdot\|$-topology and $\rho$-topology coincide on every order interval of $E$.

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J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Space, Springer-Verlag, Berlin, 1979. MR 81c:46001

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P. Meyer-Nieberg, Charakterisierung einiger topologischer und ordnungstheoretischer Eigenschaften von Banachverb$\ddot{a}$nden mit Hilfe disjunkter Folgen, Arch. Math., 24 (1973), 640-647. MR 49:5771

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-, Banach Lattices, Springer-Verlag, New York, 1991. MR 93f:46025

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H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin, 1974. MR 54:11023

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A. C. Zaanen, Riesz Spaces II, North-Holland, 1983. MR 86b:46001

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Additional Information:

Si-Kit Chung
Affiliation: Department of Mathematics, University of Hong Kong, Hong Kong
Email: schung@submaths.hku.hk

Denny Leung
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260, Singapore
Email: matlhh@nus.edu.sg

DOI: 10.1090/S0002-9939-99-05336-8
PII: S 0002-9939(99)05336-8
Keywords: Normed vector lattices, order continuous norms
Received by editor(s): May 4, 1998
Received by editor(s) in revised form: August 13, 1998
Posted: November 24, 1999
Communicated by: Dale Alspach
Copyright of article: Copyright 2000, American Mathematical Society

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