Locally Convex Properties of Baire Type Function Spaces

For an infinite Tychonoff space $X$, a nonzero countable ordinal $\alpha$ and a locally convex space $E$ over the field $\mathbb{F}$ of real or complex numbers, we denote by $B_\alpha(X,E)$ the class of Baire-$\alpha$ functions from $X$ to $E$. In terms of the space $E$ we characterize the space $B_\alpha(X,E)$ 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 $B_\alpha(X,E)$ in a strong form: $B_\alpha(X,E)$ contains a complemented subspace isomorphic to $\mathbb{F}^{\mathbb{N}}$. Applying our results to the case when $X$ is metrizable and $E=\mathbb{R}$, we show that the space $B_\alpha(X):=B_\alpha(X,\mathbb{R})$ is Baire-like (and hence barrelled), has the Grothendieck property and the Dunford-Pettis property. Further, the space $B_\alpha(X)$ is (semi-)Montel iff it is (semi-)reflexive iff it is (quasi-)complete iff $B_\alpha(X)=\mathbb{R}^X$ (for $\alpha=1$ the last equality is equivalent to $X$ of being a $Q$-space).