Dual of $\ell_\infty(X)$

Question

Given a Banach space $X$. Consider the space $\ell_\infty(X)$ which is the $\ell_\infty$-sum of countably many copies of $X$. Is there any accessible respresentation of the dual space $\ell_\infty(X)^*$? In particular, is this dual space isomorphic to the space of finitely additive $X^*$-valued measures on the powerset of $\mathbb N$ equipped with the semivariation norm?

Any references will be appreciated.

Is $E = X$ in your question? – Christopher A. Wong Nov 18 '12 at 21:23 — Christopher A. Wong, Nov 18 '12 at 21:23
Oh yes, sorry. I'll correct it. – Slavoj Žižek Nov 18 '12 at 21:36 — Slavoj Žižek, Nov 18 '12 at 21:36

score 4 · Answer 1 · answered Dec 07 '12 at 08:36

There is no good description of the dual of $\ell_\infty(X)$ as far as I know. If $X$ is finite dimensional, then the answer to your second question is yes. Otherwise, it is no, for there is no way to define an action of a finitely additive $X^*$-valued measure on $\ell_\infty(X)$ if the ball of $X$ is not compact.

Dual of $\ell_\infty(X)$

1 Answers1