Notes on free topological vector spaces

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) over bar : V (X) -> E with f = (f) over bar circle i. In this paper, the k-property, Frechet-Urysohn property, kappa-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 kappa-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. (C) 2020 Elsevier B.V. All rights reserved.