On the Semadeni derivative of Banach spaces C(K, X)
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Data
2022
Autores
Candido, Leandro 
Tipo
Artigo
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Resumo
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.