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 ℓ¹(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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-2-3,
author = {K. K. Kampoukos and S. K. Mercourakis},
title = {A new class of weakly countably determined Banach spaces},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {155-171},
zbl = {1205.46008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-2-3}
}
K. K. Kampoukos; S. K. Mercourakis. A new class of weakly countably determined Banach spaces. Fundamenta Mathematicae, Tome 209 (2010) pp. 155-171. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm208-2-3/