Re: What is the dual of l^\infty ?

In article <5g8bh150bl2d4irlvbf8intmcmdi8ck5dm@xxxxxxx>, David C.
Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:

> On 30 Aug 2005 22:26:44 -0700, tommi.hoynalanmaa@xxxxxx wrote:
>
> >What is the dual of space l\^infty defined by
> >
> > l^\infty = { (x_k)_{k=0}^\infty \subset C | sup_k |x_k| < \infty }
> >
> >and
> >
> > ||(x_k)|| = sup_k |x_k| ?
>
> The dual is very very large (much larger than l^1, for example.)
>
> Looking at it abstractly, there is a compact Hausdorff space
> K (the maximal ideal space of l^infinity regarded as a
> Banach algebra) such that l^infinity is isometrically isomorphic
> to C(K), hence the dual is the space of complex Borel measures
> on K. But this is a very large complicated K.
>

This K is known as \beta N, the Stone-Cech compactification of a
countable discrete set N. Measures on \beta N may be identified with
finitely-additive measures on N itself.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
.