Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

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

2 Answers 2

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.

share|improve this answer
add comment

Below is from Rudin, Real and Complex Analysis, Example of Banach Space Techniques.

Suppose $c_{0}$ is the subspace of $l^{\infty}(X)$ consisting of all $x$ for which $x_{i}\rightarrow 0$ as $i\rightarrow 0$. Then $c_{0}^{*}=l^{1}$. This means to every bounded linear functional $\Lambda$ on $c_{0}$ there corresponds a unique sequence $\{\delta_{i}\}$ such that $$\sum |\delta_{i}|=|\Lambda|,\Lambda x=\sum x_{i}\delta_{i}$$

The reason $(l^{\infty})^{*}\not=l^{1}$ depends on the fact that there is a nontrivial bound linear function on $l^{\infty}$ which vanishes on all of $c_{0}$.

share|improve this answer
1  
Note that in the question $X$ is a Banach space and $\ell^{\infty}(X)$ is the space of bounded functions $\mathbb{N} \to X$ so (unless I am misreading) you only seem to be addressing the case $X = \mathbb{R}$ which the OP is trying to generalize. In the case $X = \mathbb{R}$, the dual of $\ell^\infty$ can be described in terms of finitely additive measures and the OP asks whether a similar interpretation holds true for $X$ an arbitrary Banach space. –  Martin Jan 9 '13 at 7:05
    
I see. I checked Rudin again and it assumed $X$ is either $\mathbb{R}$ or $\mathbb{C}$. –  Bombyx mori Jan 9 '13 at 7:24
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.