How far is $C(\omega )$ from the other $C(K)$ spaces?
Volume 217 / 2013
Studia Mathematica 217 (2013), 123-138
MSC: Primary 46B03, 46E15; Secondary 46B25.
DOI: 10.4064/sm217-2-2
Abstract
Let us denote by $C(\alpha )$ the classical Banach space $C(K)$ when $K$ is the interval of ordinals $ [1, \alpha ]$ endowed with the order topology. In the present paper, we give an answer to a 1960 Bessaga and Pełczyński question by providing tight bounds for the Banach–Mazur distance between $C(\omega )$ and any other $C(K)$ space which is isomorphic to it. More precisely, we obtain lower bounds $L(n, k)$ and upper bounds $U(n, k)$ on $d(C(\omega ), C(\omega ^{n} k))$ such that $U(n,k)-L(n, k)<2$ for all $1 \leq n, k <\omega $.
