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

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.

> - Tommi Höynälänmaa -


************************

David C. Ullrich
.

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