Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose $X$ and $Y$ are Banach spaces with duals $X^*$ and $Y^*$. If we know $X^*$ and $Y^*$ are isomorphic, what can we say about $X$ and $Y$?

One trivial thing is that they can be isometrically embedded into the same Banach space $X^{**}$. Another is that they can be isometrically embedded into the same $C(S)$, where $S$ is a compact Hausdorff space.

But I am guessing much more can be implied about $X$ and $Y$? Am I right?


share|cite|improve this question
Isomorphic according to which topology? The norm? – Christopher A. Wong Nov 21 '12 at 18:50
$\ell_1$ has at least $\aleph_1$ many mutually nonisomorphic isometric preduals, e.g. the spaces $C(K)$ with $K$ countable (see my remarks at…). Dropping the isometric condition, you can get $2^{\aleph_0}$ many isomorphic preduals of $\ell_1$ by considering the $\mathcal{L}_\infty$ spaces of Bourgain and Delbaen (look up `Bourgain-Delbaen space'). – Philip Brooker Nov 21 '12 at 21:58
up vote 1 down vote accepted

Some things might be said beyond the ones that you mention, but probably not a lot. As an example, look at this answer, where an example is provided of separable $X$, non-separable $Y$, with $X^*=Y^*$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.