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⁎cs⁎-character. (ii) A locally convex b  -Baire-like space is metrizable if and only if it has countable cs⁎cs⁎-character. Both results extend earlier metrization theorems involving the concept of the cs⁎cs⁎-countable character. Theorem (ii) extends a theorem (Sakai) stating that the space Cp(X)Cp(X) has countable cs⁎cs⁎-character if and only if X is countable.