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An ordinal version of some applications of the classical interpolation theorem. (English) Zbl 0901.46011
Summary: Let \(E\) be a Banach space with a separable dual. Zippin’s theorem asserts that \(E\) embeds in a Banach space \(E_1\) with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pelcyzński have shown that \(E\) is a quotient of a Banach space \(E_2\) with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of \(E_1\) and \(E_2\) can be controlled by the Szlenk index of \(E\), where the Szlenk index is an ordinal index associated with a separable Banach space which provides a transfinite measure of the separability of the dual space.

MSC:
46B20 Geometry and structure of normed linear spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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