Locally Convex Properties of Baire Type Function Spaces

For an infinite Tychonoff space X, a nonzero countable ordinal α and a locally convex space E over the field F of real or complex numbers, we denote by B_α(X, E) the class of Baire-α functions from X to E. In terms of the space E we characterize the space B_α(X, E) satisfying various weak barrelledness conditions, (DF)-type properties, the Grothendieck property, or Dunford–Pettis type properties. We solve Banach–Mazur’s separable quotient problem for B_α(X, E) in a strong form: B_α(X, E) contains a complemented subspace isomorphic to F^N. Applying our results to the case when X is metrizable and E = ℝ, we show that the space B_α(X):= B_α(X, ℝ) is Baire-like (and hence barrelled), has the Grothendieck property and the Dunford-Pettis property. Further, the space B_α(X) is (semi-)Montel iff it is (semi-)reflexive iff it is (quasi-)complete iff B_α(X) = ℝ^X (for α = 1 the last equality is equivalent to X of being a Q-space).