For a locally compact Hausdorff space K and a Banach space X we denote by C₀(K,X) the space of X-valued continuous functions on K which vanish at infinity, provided with the supremum norm. Let n be a positive integer, Γ an infinite set with the discrete topology, and X a Banach space having non-trivial cotype. We first prove that if the nth derived set of K is not empty, then the Banach-Mazur distance between C₀(Γ,X) and C₀(K,X) is greater than or equal to 2n + 1. We also show that the Banach-Mazur distance between C₀(ℕ,X) and C([1,ωⁿk],X) is exactly 2n + 1, for any positive integers n and k. These results extend and provide a vector-valued version of some 1970 Cambern theorems, concerning the cases where n = 1 and X is the scalar field.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-2-3,
author = {Leandro Candido and El\'oi Medina Galego},
title = {How far is C0(G,X) with G discrete from C0(K,X) spaces?},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {151-163},
zbl = {1258.46002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-2-3}
}
Leandro Candido; Elói Medina Galego. How far is C₀(Γ,X) with Γ discrete from C₀(K,X) spaces?. Fundamenta Mathematicae, Tome 219 (2012) pp. 151-163. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-2-3/