A strengthened Orlicz–Pettis theorem via Itô–Nisio
Name
43034_2022_Article_225.pdf
Size
1.85 MB
Format
Adobe PDF
Checksum (MD5)
88298c7d0e40892ab64623e4f827799a
Author(s)
Sussman, Ethan
Date Issued
January 3, 2023
Publisher
Springer International Publishing
Citation
Annals of Functional Analysis. 2023 Jan 03;14(1):22
Version
Final published version
Abstract
Abstract
In this note, we deduce a strengthening of the Orlicz–Pettis theorem from the Itô–Nisio theorem. The argument shows that given any series in a Banach space which is not 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 Itô–Nisio theorem by Hoffmann–Jørgensen allows us to replace ‘weakly summable’ with ‘
$$\tau$$
τ
-weakly summable’ for appropriate topologies
$$\tau$$
τ
weaker than the weak topology. A treatment of the Itô–Nisio theorem for admissible
$$\tau$$
τ
is given.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s43034-022-00225-1
