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On paratopological vector spaces. (English) Zbl 1049.54029

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.

MSC:

54E35 Metric spaces, metrizability
54H11 Topological groups (topological aspects)
54H13 Topological fields, rings, etc. (topological aspects)
46A03 General theory of locally convex spaces
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