Compactness in the First Baire Class and Baire-1 Operators

Mercourakis, S., and Stamati, E.. "Compactness in the First Baire Class and Baire-1 Operators." Serdica Mathematical Journal 28.1 (2002): 1-36. <http://eudml.org/doc/11545>.

@article{Mercourakis2002,
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.},
author = {Mercourakis, S., Stamati, E.},
journal = {Serdica Mathematical Journal},
keywords = {Baire-1 Function; Baire-1 Operator; Rosenthal Compact; Rosenthal-Banach Compact; Polish Space; Angelic Space; Bounded Approximation Property; Baire-1 function; Baire-1 operator; Rosenthal compact; Rosenthal-Banach compact; Polish space; angelic space; bounded approximation property; Gulko compact},
language = {eng},
number = {1},
pages = {1-36},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Compactness in the First Baire Class and Baire-1 Operators},
url = {http://eudml.org/doc/11545},
volume = {28},
year = {2002},
}

TY - JOUR
AU - Mercourakis, S.
AU - Stamati, E.
TI - Compactness in the First Baire Class and Baire-1 Operators
JO - Serdica Mathematical Journal
PY - 2002
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 28
IS - 1
SP - 1
EP - 36
AB - 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.
LA - eng
KW - Baire-1 Function; Baire-1 Operator; Rosenthal Compact; Rosenthal-Banach Compact; Polish Space; Angelic Space; Bounded Approximation Property; Baire-1 function; Baire-1 operator; Rosenthal compact; Rosenthal-Banach compact; Polish space; angelic space; bounded approximation property; Gulko compact
UR - http://eudml.org/doc/11545
ER -