A new class of weakly countably determined Banach spaces
Volume 208 / 2010
Fundamenta Mathematicae 208 (2010), 155-171
MSC: Primary 46B20, 54H05, 03E15; Secondary 46B26.
DOI: 10.4064/fm208-2-3
Abstract
A class of Banach spaces, countably determined in their weak topology (hence, WCD spaces) is defined and studied; we call them strongly weakly countably determined (SWCD) Banach spaces. The main results are the following: (i) A separable Banach space not containing $\ell^1 (\mathbb{N})$ is SWCD if and only if it has separable dual; thus in particular, not every separable Banach space is SWCD. (ii) If $K$ is a compact space, then the space $C(K)$ is SWCD if and only if $K$ is countable.
