# Compactness in the First Baire Class and Baire-1 Operators

Serdica Mathematical Journal (2002)

- Volume: 28, Issue: 1, page 1-36
- ISSN: 1310-6600

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topMercourakis, 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 -

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