Distances to spaces of Baire one functions
type: Item
instance of: scholarly article
title: Distances to spaces of Baire one functions
zbMATH Open document ID: 1173.54007
published in: Mathematische Zeitschrift
publication date: 2009-08-27T00:00:00Z
review text: Recently several quantitative counterparts for classical results such as Krein-Šmulian, Eberlein-Šmulian, Grothendieck, etc., have been proved. The paper under review goes along the same line. In particular, the authors present quantitative versions for Rosenthal's theorem (in the space \(B_1(X)\) of Baire one functions the pointwise relatively countably compact sets are pointwise compact when \(X\) is Polish), Mazur's theorem (if \((f_n)\) is a uniformly bounded sequence of continuous functions, defined on a countably compact space \(X\), converging pointwise to a continuous function \(f\), then some convex combinations of \((f_n)\) converge to \(f\) uniformly), Srivatsa's theorem (if \(X\) is metric, \(E\) is Banach and \(f:X\to E\) is weakly continuous, then \(f\) is Baire one) and Namioka's theorem (if \(X\) is complete metric, \(K\) is compact and \(f:X\times K\to \mathbb{R}\) separately continuous, then there exists a dense \(G_\delta\) set \(D\subset X\) such that \(f\) is continuous in each point of \(D\times K\)). Let us present one result. If \(X\) is a \(\sigma-\delta\)-unfavorable space and \(K\) is a compact space, then there exists a dense \(G_\delta\) subset \(D\) of \(X\) such that the oscillation \(\mathrm{osc}(f, (y,k))\leq 6\sup_{x\in X}\mathrm{osc}(f_x) +8\sup_{k\in K}\mathrm{osc}(f^k)\) for each \((y,k)\in D\times K\).
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 46A50
zbMATH DE Number: 5598908
zbMATH Keywords: analytic spaces
zbMATH Keywords: \(\sigma\)-fragmented maps
zbMATH Keywords: Baire one functions
zbMATH Keywords: countable compactness
zbMATH Keywords: compactness
zbMATH Keywords: separate continuity
zbMATH Keywords: joint continuity
author: C. Angosto
author: B. Cascales
author: Isaac Namioka
reviewed by: A. M. Plichko
MaRDI profile type: Publication
full work available at URL: s00209-008-0412-8
OpenAlex ID: W2078707030
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DOI: 10.1007/S00209-008-0412-8
description: scientific article; zbMATH DE number 5598908