2016 seminar talk: On the choice in Rosenthal's lemma

Talk held by Damian Sobota (KGRC) at the KGRC seminar on 2016-11-10.


Rosenthal's lemma in its most basic form states that given an infinite matrix $(m_n^k)_{n,k\in\omega}$ of non-negative reals such that $\sum_{n\in\omega}m_n^k\le 1$ for every $k\in\omega$, and $\varepsilon>0$, there exists an infinite set $A\subset\omega$ such that $\sum_{n\in A,n\neq k}m_n^k\le\varepsilon$ for every $k\in A$. The lemma has numerous important applications in Banach space theory and vector measure theory — I will mention some of them during the talk (on the fly explaining and exemplifying all notions and terms).

A natural question arises — can the choice of a set $A$ in Rosenthal's lemma be somehow controlled, i.e. can $A$ be chosen from some fixed family $\mathcal{F}\subset[\omega]^\omega$? I will show that it is not possible if $\mathcal{F}$ has cardinality strictly less than $\text{cov}(\mathcal{M})$ (the covering of category). On the other hand, if $\mathcal{F}$ is a basis of a selective ultrafilter (assuming one exists), then $A$ can be chosen from $\mathcal{F}$.

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Kurt Gödel Research Center for Mathematical Logic. Währinger Straße 25, 1090 Wien, Austria. Phone +43-1-4277-50501. Last updated: 2010-12-16, 04:37.