Null-separable Banach spaces
by
Heikki Junnila
University of Helsinki

We say that a Banach space E is "null-separable" if every bounded linear mapping from E to c (T), where T is any set, has separable range. We note that if every point-finite family of weakly open subsets of E is countable, then E is null-separable. Kunen's weakly hereditarily Lindelöf C(K)-space (under CH) is a consistent example of a non-separable null-separable Banach space. Another consistent example is provided by a Banach space S constructed by Shelah under diamond: null-separability follows from the property of S that any uncountable subset A of S has a point p which belongs to the closed convex hull of the set A-{p}. In Kunen's and Shelah's spaces, every biorthogonal system is countable; we don't know, if this condition is sufficient for null-separability.

The spaces of Kunen and Shelah are "hereditarily null-separable", in the sense that every closed linear subspace is null-separable. We construct an example, under diamond, of a null-separable Banach space which contains an isomorphic copy of the non-null-separable space c (\omega₁).

We present a variety of conditions under which a null-separable Banach space E is separable; this implication holds, for instance, if E has a fundamental biorthogonal system or if E contains an isomorphic copy of l^\infty. We don't know if it is consistent that every null-separable Banach space is separable.

Date received: June 4, 2003