Szlenk and -dentability indices of the Banach spaces

Let α be an infinite ordinal and γ the unique ordinal satisfying ω ω γ ≤ α < ω ω γ + 1 . We show that the Banach space C ( [ 0 , α ] ) of all continuous scalar-valued functions on the compact ordinal interval [ 0 , α ] has Szlenk index equal to ω γ + 1 and w ∗ -dentability index equal to ω 1 + γ + 1 .