A proof of Holsztyński theorem

Abstract

For a compact Hausdorff space, we denote by C(K) the Banach space of continuous functions defined in K with values in R or C. A well known result in Banach spaces of continuous functions is the Holsztyński theorem which establishes that if C(K) is isometric to a subspace of C(S), then K is a continuous image of S. The aim of this paper is to give an alternative proof of this result for extremely regular subspaces of C(K).

Isomorphisms from Extremely Regular Subspaces of C0K into C0Cerpa-Torres1, Rincón‐Villamizar2 2019International Journal of Mathematics and Mathematical SciencesSelf Cite1010View full textAdd to dashboardCite

“…e next result is proved in ( [24], Lemma 1) for compact spaces. e proof also works in the locally compact case, so we restate it as follows.…”

“…e following result was proved for compact spaces in ( [24], eorem 2.4). By modifying slightly the proof showed in [24], we see that result also holds in the locally compact case. Theorem 6.…”