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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 $E^\star$.

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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
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.
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Thank you Pavel. – Tomás Dec 18 '12 at 14:44

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