Re: Banach space isomorphisms


In article <e2e4ke$cou$1@xxxxxxxxxxxxxxxxxxxxxxxxx>, Dan Goodman
<usenetdog@xxxxxxxxxxxxxxxxx> wrote:

If X and Y are Banach spaces, and X** and Y** denote their respective
Banach double duals, and there is an isomorphism between X** and Y**,
does this imply that there is an isomorphism between X and Y? In case
this is not true in general, it might be in the particular case I am
interested in where Y is c_{0}, the space of sequence converging to
zero, and Y** is then l^{infinity}, the space of bounded sequences.

No. If S is a countable compact metric space, then Banach space C(S)
has dual isometric to l^1, so second-dual isometric to l^infinity.
But not all C(S) spaces are isomorphic. In fact, there are aleph-1
isomorphism classes. I believe you classify them by the number
of times (possibly transfinite) you form the derived set starting with
S before you reach a finite set. (This is a countable ordinal.)
There is a topic in Banach space theory called "l^1 preduals".

--
G. A. Edgar edgar at math.ohio-state.edu
.

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