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Let $K$ be a compact, Hausdorff space and $X$ be a Banach space. By $C(K,X)$ we denote the Banach space of all continuous functions $f : K \to X$, equipped with the supremum norm: \begin{align} \|f\|_\infty = \sup_{k \in K} \|f(k)\|. \end{align}

Suppose $K$ is scattered. Is $(C(K,X))^*$, the dual space of $C(K,X)$, isomorphic to $\ell_1(I, X^*)$, for some non-empty set $I$?

I know this is true if $X = \mathbb{R}$, but despite my best efforts was unable to find a proof of this more general case.

A reference to a proof in the affirmative case, or a counter-example in the negative would be most appreciated.

Vinícius Morelli
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    This seems to follow from Singer's representation theorem: http://www.ams.org/journals/proc/1996-124-10/S0002-9939-96-03493-4/S0002-9939-96-03493-4.pdf – Tomasz Kania Aug 04 '15 at 17:55
  • Thanks for the reference. I examined the proof of the real case and was able to modify it to work in this general case, too. – Vinícius Morelli Aug 20 '15 at 10:24

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