Embedding normed linear spaces into C(X)

Abstract

It is well known that every (real or complex) normed linear space L is isometrically embeddable into C(X) for some compact Hausdorff space X. Here X is the closed unit ball of L* (the set of all continuous scalar-valued linear mappings on L) endowed with the weak* topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space X can indeed be chosen to be the Stone-Čech compactification of L* \ {0}, where L*\ {0} is endowed with the supremum norm topology. © 2017 Iranian Mathematical Society.