Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $E$ be a normed vector space with norm $\|\ \|$. Is it possible to find a equivalent norm $|\ |$ in $E$ such that $E$ and $E^\star$ are locally uniform convex spaces?

Note: Im assuming the norm $\|f\|_{E^\star}=\displaystyle\sup_{|x|\leq 1}|f(x)|$ in $E^\star$.

share|improve this question
    
No, as uniformly convex Banach spaces are reflexive. –  martini Dec 5 '12 at 20:58
    
Sorry @Martini, There is one word missing! Im gonna edit it. –  Tomás Dec 5 '12 at 21:01
add comment

1 Answer 1

up vote 1 down vote accepted
  1. It suffices to ask about $E^*$. If $E^*$ admits an equivalent locally uniformly convex (LUR) norm, then there is an equivalent norm on $E$ which makes both $E$ and $E^*$ LUR (Richard Haydon).
  2. As you probably know, every separable space has an equivalent LUR norm (Kadec renorming, 1959).
  3. $\ell_\infty$ does not admit an equivalent LUR norm. I saw this credited to Lindenstrauss ("Weakly compact sets—their topological properties and the Banach spaces they generate") but did not read the paper.
share|improve this answer
    
Thank you Pavel. –  Tomás Dec 18 '12 at 14:44
add comment

Your Answer

 
discard

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.