This is an announcement for the paper "On the minimal space problem and a new result on existence of basic sequences in quasi-Banach spaces" by Cleon S. Barroso.
Abstract: We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by Kalton. Namely, which all of its $\aleph_0$-dimensional subspaces fail to have a separating family of continuous linear functionals.
Archive classification: math.FA
Submitted from: cleonbar@mat.ufc.br
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1202.3088
or