Notes on free topological vector spaces

Abstract The free topological vector space V ( X ) over a Tychonoff space X is a pair consisting of a topological vector space V ( X ) and a continuous mapping i = i X : X → V ( X ) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f ‾ : V ( X ) → E with f = f ‾ ∘ i . In this paper, the k-property, Frechet-Urysohn property, κ-Frechet-Urysohn property and countable tightness of free topological vector space over some class of generalized metric spaces are studied. First, we mainly discuss the characterization of a space X such that V ( X ) or the third level of V ( X ) is Frechet-Urysohn or κ-Frechet-Urysohn, respectively. Then we give the characterization of a space X such that the second level of V ( X ) is of countable tightness or is a k-space, respectively.