Metrization conditions for topological vector spaces with Baire type properties

We show: (i) A Baire topological vector space is metrizable if and only if it has countable cs*-character. (ii) A locally convex b-Baire-like space is metrizable if and only if it has countable cs*-character. Both results extend earlier metrization theorems involving the concept of the cs*-countable character. Theorem (ii) extends a theorem (Sakai) stating that the space C-p(X) has countable cs*-character if and only if X is countable. (C) 2014 Elsevier B.V. All rights reserved.