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

Tom 218 / 2012

Leandro Candido, Elói Medina Galego Fundamenta Mathematicae 218 (2012), 151-163 MSC: Primary 46B03, 46E40; Secondary 46E27, 46B25 DOI: 10.4064/fm218-2-3


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.


  • Leandro CandidoDepartment of Mathematics
    IME, University of São Paulo
    Rua do Matão 1010, São Paulo, Brazil
  • Elói Medina GalegoDepartment of Mathematics
    University of São Paulo
    IME, Rua do Matão 1010, São Paulo, Brazil

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