Abstract

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

Author information

Contact details are reproduced from the original publication and may be historical.

Taras Banakh

I. Franko National University, Lviv, Ukraine
and: J. Kochanowski University, Kielce, Poland

t.o.banakh@gmail.com

Saak Gabriyelyan

Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

saak@math.bgu.ac.il

Keywords

  • Baire type function spaces
  • Baire-like
  • weak barrelledness
  • Grothendieck property
  • Dunford-Pettis property
  • quasi-DF-space
  • semi-reflexive
  • semi-Montel

Mathematics Subject Classification

46A03; 46A08, 54C35

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Suggested citation

T. Banakh, S. Gabriyelyan. “Locally Convex Properties of Baire Type Function Spaces.” Journal of Convex Analysis 28 (2021), No. 3, 803–818. https://doi.org/10.68381/jca28046

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.