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

Volume 266 / 2022

Leandro Candido Studia Mathematica 266 (2022), 225-240 MSC: Primary 46E15; Secondary 46E40. DOI: 10.4064/sm210810-9-12 Published online: 28 April 2022


The Semadeni derivative of a Banach space $X$, denoted by $\mathcal {S}(X)$, is the quotient of the space of all weak$^*$ sequentially continuous functionals in $X^{**}$ by the canonical copy of $X$. In a remarkable 1960 paper, Z. Semadeni introduced this concept in order to prove that $C([0,\omega _1])$ is not isomorphic to $C([0,\omega _1])\oplus C([0,\omega _1])$.

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 $\mathcal {S}(C(K,X))$ is isometrically isomorphic to $C(K,\mathcal {S}(X))$ for every Banach space $X$. Additionally, if $X$ is a Banach space with the Mazur property, we explicitly find the derivative of $C([0,\omega _1]^n,X)$ for each $n\geq 1$. Further we obtain an example of a nontrivial Banach space linearly isomorphic to its derivative.


  • Leandro CandidoDepartamento de Matemática
    Instituto de Ciência e Tecnologia
    Universidade Federal de São Paulo – UNIFESP
    Avenida Cesare Mansueto Giulio Lattes, 1201
    CEP 12247014, São José dos Campos, SP, Brasil

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