How far is C₀(Γ,X) with Γ discrete from C₀(K,X) spaces?

Leandro Candido, and Elói Medina Galego. "How far is C₀(Γ,X) with Γ discrete from C₀(K,X) spaces?." Fundamenta Mathematicae 218.2 (2012): 151-163. <http://eudml.org/doc/283256>.

@article{LeandroCandido2012,
abstract = {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.},
author = {Leandro Candido, Elói Medina Galego},
journal = {Fundamenta Mathematicae},
keywords = {Banach-Mazur distance; spaces of vector-valued continuous functions},
language = {eng},
number = {2},
pages = {151-163},
title = {How far is C₀(Γ,X) with Γ discrete from C₀(K,X) spaces?},
url = {http://eudml.org/doc/283256},
volume = {218},
year = {2012},
}

TY - JOUR
AU - Leandro Candido
AU - Elói Medina Galego
TI - How far is C₀(Γ,X) with Γ discrete from C₀(K,X) spaces?
JO - Fundamenta Mathematicae
PY - 2012
VL - 218
IS - 2
SP - 151
EP - 163
AB - 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.
LA - eng
KW - Banach-Mazur distance; spaces of vector-valued continuous functions
UR - http://eudml.org/doc/283256
ER -