The Szlenk index of L p ( X )

We find an optimal upper bound on the values of the weak∗ ‐dentability index Dz ( X ) in terms of the Szlenk index Sz ( X ) of a Banach space X with separable dual. Namely, if Sz ( X ) = ω α , for some α < ω 1 , and p ∈ ( 1 , ∞ ) , thenSz ( X ) ⩽ Dz ( X ) ⩽ Sz ( L p ( X ) ) ⩽ω α + 1ifαis a finite ordinal ,ω αifαis an infinite ordinal .