Optimal sequences of continuous functions converging to a Baire-1 function
type: Item
instance of: scholarly article
title: Optimal sequences of continuous functions converging to a Baire-1 function
zbMATH Open document ID: 1034.46008
author: Vassilis Kanellopoulos
author: Spiros A. Argyros
published in: Mathematische Annalen
publication date: 2003-03-27T00:00:00Z
review text: Let \(K\) be a compact metric space, \(f\) a real valued function on \(K\), and \((f_n)\) a uniformly bounded sequence of continuous functions converging pointwise to \(f\). In the topological part of this article the authors clarify the relation between the discontinuities of \(f\) and the oscillatory behavior of \((f_n)\). This is done by using the transfinite oscillations \(\text{osc}_\xi\,f\) introduced by Rosenthal and the new \(\xi\)-th variation \(\nu_\xi\) of the sequence \((f_n)\), \(\xi\) a countable ordinal. It is shown that there is a convex block sequence of \((f_n)\) which is optimal with respect to \(\text{osc}_\xi\) and \(\nu_\xi\). The motivation for this work is to give a proof of Rosenthal's \(c_0\)-index conjecture. For a bounded function \(f\) on \(K\) which is not in \(D(K)\), the set of differences of bounded semicontinuous functions, it is known that there is a least countable ordinal \(\xi\) such that \(\text{osc}_\xi\,f\) is unbounded. This ordinal \(\xi\) is the non-\(D\) index \(r_{ND}(f)\) and measures the distance to \(D(K)\) in the sense that \(r_{ND}(f)\) is small if \(f\) is ``far from \(D\)''. The conjecture then said, that for a separable Banach space \(X\) not containing \(c_0\) and \(K=(B_{X^*},\omega^*)\) there is a countable ordinal \(\xi\) such that \(r_{ND}(x^{**}| K)\leq\xi\) for all \(x^{**}\in X^{**}\setminus X\). This is a strong quantitative improvement of the result that \(X\) contains \(c_0\) iff there is \(x^{**}\in X^{**}\setminus X\) which is in \(D(K)\). The proof is long and intricate. An introduction to the topic is [\textit{S. A. Argyros, G. Godefroy} and \textit{H. P. Rosenthal} [Descriptive set theory and Banach spaces, in: Handbook of the geometry of Banach Spaces. Vol. 2. Amsterdam: North-Holland, 1007--1069 (2003; Zbl 1121.46008)], where the present article is already integrated.
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 1889189
zbMATH Keywords: Rosenthal's \(c_0\)-index
reviewed by: Peter Harmand
MaRDI profile type: Publication
full work available at URL: s00208-002-0354-0
OpenAlex ID: W1969538202
DOI: 10.1007/S00208-002-0354-0
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description: scientific article; zbMATH DE number 1889189