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

Nawrocki, M.

doi:10.1090/S0002-9939-1987-0908655-3

On the Orlicz-Pettis property in nonlocally convex $F$-spaces
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Proc. Amer. Math. Soc. 101 (1987), 492-496 Request permission

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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