论文标题
通过itô-nisio加强了Orlicz-Pettis定理
A strengthened Orlicz-Pettis theorem via Itô-Nisio
论文作者
论文摘要
在本说明中，我们从itô-nisio定理中推断出orlicz-pettis定理的加强。该论点表明，鉴于Banach空间中的任何序列，这是不可总结的（或更一般无条件的总结），我们可以构建一个（粗粒）的子层，并具有该属性 - 在某些适当的“几乎所有”概念下，几乎所有其他的子层都无法忽略。此外，霍夫曼 - 乔根森（Hoffmann-Jorgensen）对itô-nisio定理的加强使我们能够用“ $τ$ - 非常易于总结”代替“弱总结”，以使适当的拓扑结构$τ$比弱拓扑较弱。给出了对可允许$τ$的ITô-nisio定理的处理。
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 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 Itô-Nisio theorem by Hoffmann-Jorgensen allows us to replace `weakly summable' with `$τ$-weakly summable' for appropriate topologies $τ$ weaker than the weak topology. A treatment of the Itô-Nisio theorem for admissible $τ$ is given.
