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

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