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Smoothness and estimates of sequences in Banach spaces. (English) Zbl 0823.46013

Summary: An upper bound for the order of smoothness of bump functions in Banach spaces without copy of \(c_ 0\) is found in terms of lower and upper estimates of their sequences. It is also shown that every \(C^ \infty\)- smooth Banach space with symmetric basis either contains \(c_ 0\) or is isomorphic to \(\ell_{2n}\) for some integer \(n\).

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B25 Classical Banach spaces in the general theory
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