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We show that each first countable paratopological vector space X has a compatible translation invariant quasi-metric such that the open balls are convex whenever X is a pseudoconvex vector space. We introduce the notions of a right-bounded subset and of a right-precompact subset of a
paratopological vector space X and prove that X is quasi-normable if and only if the origin has a convex and right-bounded neighborhood. Duality in this context is also
discussed. Furthermore, it is shown that the bicompletion of any paratopological vector space (respectively, of any quasi-metric
vector space) admits the structure of a paratopological vector space (respectively, of a quasi-metric vector space). Finally,
paratopological vector spaces of finite dimension are considered.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
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