A note on the Schur and Phillips lemmas

It is well-known that every weakly convergent sequence in ${\mathrm{\ell }}_1$ 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 ${\mathrm{\ell }}_{\mathrm{\infty }}^{\mathrm{\prime }}$ converges pointwise on $\{0,1{\}}^{\mathbb{N}}$ to $0$, then its ${\mathrm{\ell }}_1$-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.