An ordinal version of some applications of the classical interpolation theorem
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. Pełczyński 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.