Dual spaces of uniformly convex Banach spaces

Question

I am interested in the dual spaces of uniformly convex Banach spaces. Given a uniformly convex Banach space $X$, can anything be said about uniform convexity of its dual space $X^*$? Or given a uniformly convex dual space $X^*$, can anything be said about the uniform convexity of $X$?

I would like links to any papers discussing these points, or any counterexamples. Thank you.

Answer 1

A Banach space is uniformly convex if and only if its dual is uniformly smooth. Also, since any uniformly convex Banach space is reflexive, this goes both ways, and any uniformly smooth Banach space is also reflexive. This can all be found on the Wikipedia articles for these properties, and they contain references at the bottom.

EDIT: Because uniformly convex and uniformly smooth are independent, we can find Banach spaces that are uniformly convex but which have a dual space that is not, and vise versa. Same for uniformly smooth.

Example: The norm $\|\cdot\|_1+\|\cdot\|_2$ is uniformly convex, but not uniformly smooth if $2\leq\dim X<\infty$. Hence its dual is uniformly smooth, but not uniformly convex.

Answer 2

This question has the merit to deserve more attention.

Definition A Banach space is uniformly convexifiable (uniformly smoothable) if it can be given an equivalent norm that makes it uniformly convex(uniformly smooth).

Theorem[Per Enflo] See P. Enflo Corollary 4. A Banach space is uniformly smaoothable if and only if its dual is uniformly smoothable.

Corollary A consequence of P. Enflo Corollary 4 implies: A Banach space is uniformly convexfiable if and only if its dual is uniformly convexifiable.

Another consequence Note that uniform convexity may not be transferred from the space to its dual. However if $X$ is uniformly convex then its dual $X^*$ is uniformly convexifiable(vice-versa).