How far is $C_{0}(\varGamma, X)$ with $\varGamma$ discrete from $C_{0}(K, X)$ spaces?
Volume 218 / 2012
Abstract
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.
