Optimal sequences of continuous functions converging to a Baire-1 function
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.
scientific article; zbMATH DE number 1889189