How far is $C_{0}(\varGamma, X)$ with $\varGamma$ discrete from $C_{0}(K, X)$ spaces?

Leandro Candido ¹   ; Elói Medina Galego ²

¹ Department of Mathematics IME, University of São Paulo Rua do Matão 1010, São Paulo, Brazil
² Department of Mathematics University of São Paulo IME, Rua do Matão 1010, São Paulo, Brazil

Fundamenta Mathematicae, Tome 218 (2012) no. 2, pp. 151-163

Résumé

For a locally compact Hausdorff space $K$ and a Banach space $X$ we denote by $C_{0}(K, X)$ the space of $X$-valued continuous functions on $K$ which vanish at infinity, provided with the supremum norm. Let $n$ be a positive integer, $\varGamma$ an infinite set with the discrete topology, and $X$ a Banach space having non-trivial cotype. We first prove that if the $n$th derived set of $K$ is not empty, then the Banach–Mazur distance between $C_{0}(\varGamma, X)$ and $C_{0}(K, X)$ is greater than or equal to $2n+1$. We also show that the Banach–Mazur distance between $C_{0}(\mathbb N, X)$ and $C([1, \omega^{n} k], X)$ 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.

Détail

DOI : 10.4064/fm218-2-3

Keywords: locally compact hausdorff space banach space denote space x valued continuous functions which vanish infinity provided supremum norm positive integer vargamma infinite set discrete topology banach space having non trivial cotype first prove nth derived set empty banach mazur distance between vargamma greater equal banach mazur distance between mathbb omega exactly positive integers these results extend provide vector valued version cambern theorems concerning cases where scalar field