On the Semadeni derivative of Banach spaces C(K, X)

Abstract

The Semadeni derivative of a Banach space X, denoted by S(X), is the quotient of the space of all weak* sequentially continuous functionals in X** by the canon- ical copy of X. In a remarkable 1960 paper, Z. Semadeni introduced this concept in order to prove that C([0, ω 1 ]) is not isomorphic to C([0, ω 1 ]) ⊕ C([0, ω 1 ]). Here we investigate this concept in the context of C(K, X) spaces. In our main result, we prove that if K is a Hausdorff compactum of countable height, then S(C(K, X)) is isometrically isomorphic to C(K, S(X)) for every Banach space X. Additionally, if X is a Banach space with the Mazur property, we explicitly find the derivative of C([0, ω 1 ] n , X) for each n ≥ 1. Further we obtain an example of a nontrivial Banach space linearly isomorphic to its derivative.