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

I'm trying to find an example of a space $V$ which is strictly convex, but has a dual space $V^*$ which is not strictly smooth. Any help please?

share|cite|improve this question
Actually, I don't know what strictly smooth means. I know that smooth means the uniqueness of the norming functional for every nonzero element. – user31373 Sep 14 '12 at 3:25
up vote 2 down vote accepted

As any separable space, $\ell_1$ admits an equivalent strictly convex norm. Indeed, since the identity map from $\ell_1$ to $\ell_2$ is bounded, we can take $\|x\|_1+\|x\|_2$ as such a norm. The dual of this strictly convex space is isomorphic to $\ell_\infty$. But M.M. Day proved in 1955 (see Theorem 9) that $\ell_\infty$ does not admit any equivalent smooth norm.

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.