Embedding normed linear spaces into C(X)

TL;DR

This paper demonstrates that any normed linear space can be isometrically embedded into a space of continuous functions on a specific compactification of its dual space, refining the classical embedding approach.

Contribution

It shows that the embedding space can be chosen as the Stone-Cech compactification of the dual space minus zero, with the supremum norm topology.

Findings

Embedding space is the Stone-Cech compactification of the dual space minus zero.

The compactification is constructed using the supremum norm topology.

The result generalizes classical embedding theorems for normed spaces.

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