Distances to spaces of Baire one functions
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 \((y,k)\in D\times K\).
scientific article; zbMATH DE number 5598908
scientific article