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

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.