A Korovkin theorem for abstract Lebesgue spaces
Let \(E\) be an AL-space (a Kakutani abstract L-space) with a generalized weak unit \(\{e_n\}\). The theorem proved here is a little stronger then the following. Theorem: Let \(\{T_i\}\) be a sequence of norm one operators defined on \(E\). Let \(L=\{ f\in E: T_if\) converges weakly to \(f\}\), and \(N=\{f\in E: T_if\) converges in norm to \(f\}\). If, for each \(i\), \(T_ie_n\) converges in norm to \(e_i\), then (i) \(N=L\), and (ii) \(N\) is a lattice. Even when restricted to basic spaces such as \(E=L_1[0,1]\), this theorem generalizes stated results in the literature. \{Reviewer's comment: The author compares a result here with the reviewer's work [J. Approximation Theory 1, 381-390 (1968; Zbl 0167.12904)]. To the reviewer's chagrin, it has repeatedly been pointed out that the theorem there stating that \(\{1, x, x^2\}\) is a Korovkin set is true when the function \(x^2\) is deleted from the set. No changes to the proof are required\}.
scientific article; zbMATH DE number 1417549