How does the distortion of linear embedding of [formula omitted] into [formula omitted] spaces depend on the height of [formula omitted

Abstract

Let C0(K) denote the space of all continuous scalar-valued functions defined on the locally compact Hausdorff space K which vanish at infinity, provided with the supremum norm. Let Γ be an infinite set endowed with discrete topology and X a Banach space. We denote by C0(Γ,X) the Banach space of X-valued functions defined on Γ which vanish at infinity, provided with the supremum norm. In this paper, we prove that, if X has non-trivial cotype and there exists a linear isomorphism T from C0(K) into C0(Γ,X), then K has finite height ht(K), and the distortion ‖T‖‖T−1‖ is greater than or equal to 2ht(K)−1. The statement of this theorem is optimal and improves a 1970 result of Gordon.