the double dual of "little l one" sequence space

Question

I remember a professor remarking a while back that the double dual of the sequence space $l_1^{\infty}(\mathbb{R})$ is a very complicated space. I understand it must contain a copy of the original space. Past that how much is understood about the double dual? I am looking for sources on this topic, books or papers included.

Answer 1

The space about whose double dual you are asking is usually denoted $\ell_1$. It is a very well-understood space, and so is its dual $\ell_1^*=\ell_\infty$. Many of the properties of $\ell_1^{**}=\ell_\infty^*$ are therefore inherited by virtue of it being the dual of a well-understood space. For instance, it is well-known that every separable space embeds into $\ell_\infty$.

A nice introduction to $\ell_\infty^*$ is given in Diestel's book Sequences and Series in Banach Spaces, p76cc. A very nice characterization is given there as follows.

Let $\Sigma$ be a $\sigma$-field of subsets of a set $\Omega$, and denote by $B(\Sigma)$ the set of bounded, $\Sigma$-measurable scalar-valued functions on $\Sigma$, and note that $B(\Sigma)$ is a Banach space when endowed with the sup norm $\|\cdot\|_\infty$. Also denote by $ba(\Sigma)$ the set of all finitely additive scalar-valued signed measures with bounded total variation, and note that $ba(\Sigma)$ is also a Banach space, as long as it is endowed with the total variation norm $\|\cdot\|_1$. We can now make the isometric identification $B(\Sigma)^*=ba(\Sigma)$ via the action $\mu(f)=\int f\,d\mu$ for all $\mu\in ba(\Sigma)$ and $f\in B(\Sigma)$. If $\Omega=\mathbb{N}$ and $\Sigma$ is the counting measure then $B(\Sigma)=\ell_\infty$, and hence $\ell_\infty^*$ is precisely the space of finitely additive signed measures on $\mathbb{N}$ with bounded total variation, denoted $ba$ for short. More information on this identification can be found in Dunford-Schwartz's Linear Operators I (on p296, it seems).

Note that, according to Diestel, you should read this paper to get a better understanding of the space $ba$. However, I have not read it myself, so beware : )

There is another natural identification, due to the fact that $\ell_\infty=C(\beta\mathbb{N})$, where $\beta\mathbb{N}$ denotes the Stone-Cech compactification of $\mathbb{N}$. Thus, $\ell_\infty^*$ can be identified with $C(\beta\mathbb{N})^*$, which in turn is identifiable with the space of regular Borel measures on $\beta\mathbb{N}$. See chapter 15 of Carother's A Short Course In Banach Space Theory on this, especially remark 2 on p152.

Also, note that this question has been asked before on MO, here. If you're wondering about the title of the question, it's because $\ell_\infty\cong L_\infty[0,1]$, and so $\ell_\infty^*\cong L_\infty[0,1]^*$ (isomorphically, not isometrically).