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ON FAMILIES OF SUBSETS OF NATURAL NUMBERS DECIDING THE NORM CONVERGENCE IN l1.
- Source :
-
Proceedings of the American Mathematical Society . Apr2018, Vol. 146 Issue 4, p1673-1680. 8p. - Publication Year :
- 2018
-
Abstract
- 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]
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 146
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 127728573
- Full Text :
- https://doi.org/10.1090/proc/13861