## Abstract

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).

Original language | English |
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Journal | Journal of Convex Analysis |

Volume | 28 |

Issue number | 3 |

State | Published - 1 Jan 2021 |

## Keywords

- (quasi-)(DF)-space
- (semi-)Montel
- (semi-)reflexive
- Baire type function spaces
- Baire-like
- Dunford-Pettis property
- Grothendieck property
- Weak barrelledness