Alegre, C.; Romaguera, S. On paratopological vector spaces. (English) Zbl 1049.54029 Acta Math. Hung. 101, No. 3, 237-261 (2003). Summary: 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. Cited in 4 Documents MSC: 54E35 Metric spaces, metrizability 54H11 Topological groups (topological aspects) 54H13 Topological fields, rings, etc. (topological aspects) 46A03 General theory of locally convex spaces Keywords:pseudoconvex; paratopological group; translation invariant quasi-metric; right-bounded; quasi-norm; continuous linear map; bicompletion PDFBibTeX XMLCite \textit{C. Alegre} and \textit{S. Romaguera}, Acta Math. Hung. 101, No. 3, 237--261 (2003; Zbl 1049.54029) Full Text: DOI