We prove that if $X$ is a quasi-normed space which possesses an infinitecountable dimensional subspace with a separating dual, then it admits astrictly weaker Hausdorff vector topology. Such a topology is constructedexplicitly. As an immediate consequence, we obtain an improvement of awell-known result of Kalton-Shapiro and Drewnowski by showing that aquasi-Banach space contains a basic sequence if and only if it contains aninfinite countable dimensional subspace whose dual is separating. We also usethis result to highlight a new feature of the minimal quasi-Banach spaceconstructed by Kalton. Namely, which all of its $\aleph_0$-dimensionalsubspaces fail to have a separating family of continuous linear functionals.
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机译：我们证明，如果$ X $是一个准赋范空间，它具有无限的维子空间和一个分离对偶，则它可以接受严格较弱的Hausdorff向量拓扑。明确构造了这种拓扑。作为直接的结果，我们通过证明拟贝纳赫空间包含且仅当它包含对偶是分开的无限可数维子空间时，才获得已知的Kalton-Shapiro和Drewnowski结果的改进。我们还使用该结果来强调由Kalton构造的最小拟Banach空间的新特征。即，其所有$ \ aleph_0 $维子空间都没有单独的连续线性函数族。
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