ON FAMILIES OF SUBSETS OF NATURAL NUMBERS DECIDING THE NORM CONVERGENCE IN l1.

The classical Schur theorem asserts that the weak convergence and the norm convergence in the Banach space l1 coincide. In this paper we study complexity and cardinality of subfamilies F of ℘(ω) such that a sequence 〈xn : n ∈ ω〉 ⊆ l1 is norm convergent whenever limn→∞∑j∈A xn(j) = 0 for every A ∈ F. We call such families Schur and prove that they cannot have cardinality less than the pseudo-intersection number p. On the other hand, we also show that every non-meager subset of the Cantor space 2ω is a Schur family when thought of as a subset of ℘(ω), implying that the minimal size of a Schur family is bounded from above by non(M), the uniformity number of the ideal of meager subsets of 2ω. [ABSTRACT FROM AUTHOR]