Baire-1 Function Baire-1 Operator Rosenthal Compact Rosenthal-Banach Compact Polish Space Angelic Space Bounded Approximation Property
Issue Date:
2002
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 28, No 1, (2002), 1p-36p
Abstract:
For a polish space M and a Banach space E let B1 (M, E)
be the space of first Baire class functions from M to E, endowed with the
pointwise weak topology. We study the compact subsets of B1 (M, E) and
show that the fundamental results proved by Rosenthal, Bourgain, Fremlin,
Talagrand and Godefroy, in case E = R, also hold true in the general
case. For instance: a subset of B1 (M, E) is compact iff it is sequentially
(resp. countably) compact, the convex hull of a compact bounded subset of
B1 (M, E) is relatively compact, etc. We also show that our class includes
Gulko compact.
In the second part of the paper we examine under which conditions a
bounded linear operator T : X ∗ → Y so that T |BX ∗ : (BX ∗ , w∗ ) → Y is a
Baire-1 function, is a pointwise limit of a sequence (Tn ) of operators with
T |BX ∗ : (BX ∗ , w∗ ) → (Y, · ) continuous for all n ∈ N. Our results in this
case are connected with classical results of Choquet, Odell and Rosenthal.
