A note on the Schur and Phillips lemmas

TL;DR

This paper explores the relationship between Schur's and Phillips' lemmas, demonstrating how a category-based version of Schur's lemma can extend Phillips' lemma to broader subsets of the Cantor space.

Contribution

It introduces a new approach using the second category version of Schur's lemma to generalize Phillips' lemma to larger subsets with interpolation properties.

Findings

Second category version of Schur's lemma proved with a short proof

Phillips' lemma extended to subsets of the Cantor space containing all finite sets

Method allows broader applications of convergence results in functional analysis

Abstract

It is well-known that every weakly convergent sequence in $\ell_1$ is convergent in the norm topology (Schur's lemma). Phillips' lemma asserts even more strongly that if a sequence $(\mu_n)_{n\in\mathbb N}$ in $\ell_\infty'$ converges pointwise on $\{0,1\}^\mathbb N$ to $0$, then its $\ell_1$-projection converges in norm to $0$. In this note we show how the second category version of Schur's lemma, for which a short proof is included, can be used to replace in Phillips' lemma $\{0,1\}^\mathbb N$ by any of its subsets which contains all finite sets and having some kind of interpolation property for finite sets.