Embedding normed linear spaces into C(X)

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-Cech compactification of \(L^*\setminus\{0\}\), where \(L^*\setminus\{0\}\) is endowed with the supremum norm topology.