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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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