On the Orlicz-Pettis property in nonlocally convex -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 -spaces with separating duals, the Orlicz-Pettis theorem fails for , and some other nonseparable -spaces of harmonic functions. In this paper we give new, much simpler examples of -spaces for which the Orlicz-Pettis theorem fails; namely weak- sequence spaces for . We observe that if then the space 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0908655-3

Keywords:
The Orlicz-Pettis theorem,
weak- spaces,
Orlicz spaces

Article copyright:
© Copyright 1987
American Mathematical Society