How far is \(C_{0}(\varGamma, X)\) with \(\varGamma\) discrete from \(C_{0}(K, X)\) spaces? (Q2900984)

For the sequence spaces \(c_0\) and \(c\), an interesting result due to \textit{M. Cambern} [Math. Ann. 188, 23--25 (1970; Zbl 0188.19101)] is that the Banach-Mazur distance \(d(c_0,c) = 3\). In this paper, the authors are interested in computing upper and lower bounds for the Banach-Mazur distance, for isomorphic spaces of continuous functions taking values in a Banach space \(X\) having non-trivial cotype.NEWLINENEWLINEFor an infinite discrete set \(\Gamma\), and for a locally compact Hausdorff space \(K\) such that the derived set \(K^{(n)} \neq \emptyset\) (for \(1 \leq n < \omega\)), the authors obtain that \(d (C_0(\Gamma,X),C_0(K,X)) \geq 2n+1\). For \(n,k\) in the above range and for any Banach space \(X\), the authors obtain that \(d(C_0(\mathbb N,X),C([1,\omega^n k],X)) \leq 2n+1\).