On the existence of overcomplete sets in some classical nonseparable Banach spaces

摘要

For a Banach space Xits subset Y subset of X is called overcomplete if vertical bar Y vertical bar = dens(X) and Z is linearly dense in X for every Z. Ywith vertical bar Z vertical bar = vertical bar Y vertical bar. In the context of nonseparable Banach spaces this notion was introduced recently by T. Russo and J. Somaglia but overcomplete sets have been considered in separable Banach spaces since the 1950ties. We prove some absolute and consistency results concerning the existence and the nonexistence of overcomplete sets in some classical nonseparable Banach spaces. For example: c(0)(omega(1)), C([0,omega(1)]), L-1({0, 1}omega(1)), l(p)(omega(1)), L-p({0, 1}omega(1)) for p is an element of (1, infinity) or in general WLD Banach spaces of density omega(1) admit overcomplete sets (in ZFC). The spaces l(infinity), l(infinity)/c(0), spaces of the form C(K) for Kextremally disconnected, superspaces of l(1)(omega(1)) of density omega(1) do not admit overcomplete sets (in ZFC). Whether the Johnson-Lindenstrauss space generated in l(infinity) by c(0) and the characteristic functions of elements of an almost disjoint family of subsets of N of cardinality omega(1) admits an overcomplete set is undecidable. The same refers to all nonseparable Banach spaces with the dual balls of density omega(1)which are separable in the weak* topology. The results proved refer to wider classes of Banach spaces but several natural open questions remain open. (C) 2021 Elsevier Inc. All rights reserved.