TITLE: The Duals of $l^\infty$ and $L^{\infty}$
QUESTION [79 upvotes]: Can we identify the dual space of $l^\infty$ with another "natural space"?  If the answer is yes, what can we say about $L^\infty$?
By the dual space I mean the space of all continuous linear functionals.

REPLY [35 votes]: This is perhaps not such a nice characterization as finitely additive measures on $\mathbb N$, but it might be worth mentioning. (You can look it as follows: We obtain  nicer measures - $\sigma$-additive instead of finitely additive - on a more complicated space - $\beta\mathbb N$ instead of $\mathbb N$.)
The space $\ell_\infty$ is isometrically isomorphic to $C(\beta\mathbb N)$, hence the dual is isomorphic to $C^*(\beta\mathbb N)$.
More details about the correspondence between $\ell_\infty$ and the Stone-Cech compactification of integers can be found at wikipedia or in chapter 15 of Carothers' book A short course on Banach space theory.
Now, $C^*(\beta\mathbb N)$ is the space of regular Borel measures on $\beta\mathbb N$ by Riesz representation theorem.
In fact, Carothers goes the other way round. First, the dual of $C^*(\beta\mathbb N)$ is described via finitely additive measures. And then he uses this to prove Riesz representation theorem for compact spaces. (This proof of Riesz' representation theorem  is due to Garling.)