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?
Example of strictly convex space with not strictly smooth dual
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convex-analysis
banach-spaces
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0Actually, I don't know what *strictly smooth* means. I know that *smooth* means the uniqueness of the norming functional for every nonzero element. – 2012-09-14
1 Answers
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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.