Locally Convex Properties of Baire Type Function Spaces

For an infinite Tychonoff space

, a nonzero countable ordinal

\alpha α

and a locally convex space

E E

over the field

\mathbb{F} F

of real or complex numbers, we denote by

B_\alpha(X,E) B α ( X , E )

the class of Baire-

\alpha α

functions from

X X

to

E E

. In terms of the space

E E

we characterize the space

B_\alpha(X,E) B α ( X , E )

satisfying various weak barrelledness conditions,

(DF) ( D F )

-type properties, the Grothendieck property, or Dunford-Pettis type properties. We solve Banach-Mazur's separable quotient problem for

B_\alpha(X,E) B α ( X , E )

in a strong form:

B_\alpha(X,E) B α ( X , E )

contains a complemented subspace isomorphic to

\mathbb{F}^{\mathbb{N}} F N

. Applying our results to the case when

X X

is metrizable and

E=\mathbb{R} E = R

, we show that the space

B_\alpha(X):=B_\alpha(X,\mathbb{R}) 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_\alpha(X) B α ( X )

is (semi-)Montel iff it is (semi-)reflexive iff it is (quasi-)complete iff

B_\alpha(X)=\mathbb{R}^X B α ( X ) = R X

(for

\alpha=1 α = 1

the last equality is equivalent to

X X

of being a

Q Q

-space).