A strengthened Orlicz-Pettis theorem via It\^o-Nisio

TL;DR

This paper strengthens the Orlicz-Pettis theorem by leveraging the Itô-Nisio theorem, showing that non-summable series in Banach spaces contain subseries with specific non-summability properties under various topologies.

Contribution

It introduces a strengthened version of the Orlicz-Pettis theorem using the Itô-Nisio theorem, extending results to $ au$-weak summability for weaker topologies.

Findings

Construction of subseries with non-summability properties

Extension to $ au$-weak summability under weaker topologies

A new treatment of the Itô-Nisio theorem for admissible $ au$

Abstract

In this note we deduce a strengthening of the Orlicz-Pettis theorem from the It\^o-Nisio theorem. The argument shows that given any series in a Banach space which isn't summable (or more generally unconditionally summable), we can construct a (coarse-grained) subseries with the property that -- under some appropriate notion of "almost all" -- almost all further subseries thereof fail to be weakly summable. Moreover, a strengthening of the It\^o-Nisio theorem by Hoffmann-Jorgensen allows us to replace `weakly summable' with `$\tau$-weakly summable' for appropriate topologies $\tau$ weaker than the weak topology. A treatment of the It\^o-Nisio theorem for admissible $\tau$ is given.