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 $L^{\ast }$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak${}^{\ast }$ 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-Cech compactification of $L^{\ast }\setminus \{0\}$, where $L^{\ast }\setminus \{0\}$ is endowed with the supremum norm topology.

4 pages