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