On a Theorem of Arvanitakis

Abstract

Arvanitakis [2] established recently a theorem which is a common generalization of Michael's convex selection theorem [11] and Dugundji's extension theorem [7]. In this note we provide a short proof of a more general version of Arvanitakis' result.1991 Mathematics Subject Classification. Primary 54C60, 46E40; Secondary 28B20.

On a simultaneous selection theorem

“…Proof. Our proof is the same as that of [22,Theorem 1.2]. We present it here for the sake of completeness.…”

“…Let C b (X, E) denote the set of all bounded continuous mappings from X to E. Valov [22,Theorem 1.2] proved that the assumption in Theorem 1.1 that X is a k-space can be dropped if C(Y, E) and C(X, E) are replaced with C b (Y, E) and C b (X, E), respectively, or if E is a Banach space. His proof is based on the existence of a barycenter map introduced by Banakh [2] and that of a perfect Milyutin mapping due to Repovš, Semenov and Shchepin [18].…”