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Fragmentability and Sigma‐Fragmentability of Banach Spaces
Author(s) -
Kenderov Petar S.,
Moors Warren B.
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079900753x
Subject(s) - mathematics , banach space , unit sphere , pointwise convergence , norm (philosophy) , topology (electrical circuits) , pointwise , ball (mathematics) , sigma , weak topology (polar topology) , pure mathematics , discrete mathematics , combinatorics , topological space , general topology , mathematical analysis , extension topology , physics , computer science , approx , quantum mechanics , political science , law , operating system
It is proved that, for a Banach space X , the following properties are equivalent: (a) ( X , weak ) is fragmentable by a metric d (., .) that majorizes the norm topology (that is, the topology generated by d contains the norm topology), (b) ( X , weak ) is fragmentable by a metric d (., .) that majorizes the weak topology, and (c) ( X , weak ) is sigma‐fragmentable by the norm. The paper gives a game characterization of these equivalent properties and uses it to show that a large class of Banach spaces (including the spaces that are Čech‐analytic in their weak topology) enjoy the properties (a)–(c). In particular, if the unit ball B of X is a Borel subset of the second dual ball ( B **, weak *) then X possesses the properties (a)–(c). This provides an alternative approach to the proofs of some of the results of J. E. Jayne, I. Namioka and C. A. Rogers on sigma‐fragmentability of Banach spaces. It is also proved that C ( T ), with the topology p of pointwise convergence, is sigma‐fragmentable by the norm whenever T =∪ i ⩾1 T i and ( C ( T i ), p ), i ⩾1, is sigma‐fragmentable by the norm. This answers a question of R. Haydon. Finally, a simple proof is given that l ∞ does not have any of the properties (a)–(c) and that ( l ∞ (Γ), weak ) is not sigma‐fragmentable by any metric, provided that the cardinality of Γ is bigger than the continuum.