Metric Compactness Criteria Involving Sequences of Mappings and a Proof of the Ascoli–Arzelà Theorem with the use of Bernstein Polynomials

Abstract

AbstractWe establish inter alia a compactness criterion in metric spaces involving a sequence of completely continuous mappings, which is continuously convergent, in the sense of H. Hahn, to the identity mapping. For Banach spaces, the linear version of that result coincides with the compactness theorem due to Mazur. We also present a new proof of the Ascoli–Arzelà theorem, in which we use the above compactness criterion applied to the sequence of Bernstein operators.