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 $\ell_\infty$. Hence, every separable space is a quotient of $\ell_\infty^*$. Etc., etc. 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][1] 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][2]. If you're wondering about the title of the question, it's because $\ell_\infty=L_\infty$, and so $\ell_\infty^*=L_\infty^*$. [1]: http://www.ams.org/journals/tran/1952-072-01/S0002-9947-1952-0045194-X/S0002-9947-1952-0045194-X.pdf [2]: http://math.stackexchange.com/questions/47395/the-duals-of-l-infty-and-l-infty