For eagles only: probably the most difficult proof of the Arzela-Ascoli theorem - via the Stone-Cech compactification

Abstract

Let C(K) denote the Banach space of all (real or complex) continuous functions on a compact Hausdorff space K. We present a novel point of view on the classical Arzela-Ascoli theorem: For every pointwise bounded and equicontinuous subset F of C(K) there is a continuous mapping J :βF → C(K), where F denotes the Stone-Cech compactication of F, such that F ⊂ J(βF); hence the closure of F is compact.

Mathematics Subject Classication (2010): Primary 54D30; Secondary 46E15, 46B85.

Key words: Arzela-Ascoli theorem, Stone-Cech compactication.