On the Semadeni derivative of Banach spaces $C(K,X)$
Volume 266 / 2022
Abstract
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.
