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How far is \(C(\omega)\) from the other \(C(K)\) spaces? (English) Zbl 1288.46013

Summary: Denote by \(C(\alpha )\) the classical Banach space \(C(K)\) where \(K\) is the interval of ordinals \( [1, \alpha ]\) endowed with the order topology. In the present paper, we give an answer to a question posed in 1960 by C. Bessaga and A. Pełczyński [Stud. Math. 19, 53–62 (1960; Zbl 0094.30303)] 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 \(1 \leq n, k <\omega \).

MSC:

46B25 Classical Banach spaces in the general theory
46B03 Isomorphic theory (including renorming) of Banach spaces
46E15 Banach spaces of continuous, differentiable or analytic functions

Citations:

Zbl 0094.30303
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