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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the Orlicz-Pettis property in nonlocally convex $ F$-spaces


Author: M. Nawrocki
Journal: Proc. Amer. Math. Soc. 101 (1987), 492-496
MSC: Primary 46A06
MathSciNet review: 908655
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Abstract: Recently, J. H. Shapiro showed that, contrary to the case of separable $ F$-spaces with separating duals, the Orlicz-Pettis theorem fails for $ {h_p},0 < p < 1$, and some other nonseparable $ F$-spaces of harmonic functions. In this paper we give new, much simpler examples of $ F$-spaces for which the Orlicz-Pettis theorem fails; namely weak-$ {L_p}$ sequence spaces $ l\left( {p,\infty } \right)$ for $ 0 < p \leq 1$. We observe that if $ 0 < p < 1$ then the space $ l\left( {p,\infty } \right)$ is nonseparable but separable with respect to its weak topology. Moreover, we show that the Orlicz-Pettis theorem holds for every Orlicz sequence space (even nonseparable).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0908655-3
PII: S 0002-9939(1987)0908655-3
Keywords: The Orlicz-Pettis theorem, weak-$ {L_p}$ spaces, Orlicz spaces
Article copyright: © Copyright 1987 American Mathematical Society