I found this beautiful theorem due to Milman and Pettis:
Every uniformly convex Banach space is reflexive.
I think it's a remarkable statement, since uniformly convexity is a geometric property of the norm and therefore need not to be true for an equivalent norm.
While reflexivity is a topological statement and therefore a reflexive space remains reflexive for an equivalent norm.
My first question is:
Could someone give an example of a space with two equivalent norms such that the space is uniformly convex for just one of the two norms.
And my second question is:
Could someone give me an example of a reflexive space that admits no uniformly convex equivalent norm.
Thank you in advance for answering my questions!
Best Answer
The answer to the first question is probably disappointingly simple: Take $\mathbb{R}^2$, once with the Euclidean norm $\|(x,y)\|_2 = (|x|^2+|y|^2)^{1/2}$ and once with the $\ell^1$-norm $\|(x,y)\|_1 = |x| + |y|$ for example. The norms are equivalent, the Euclidean norm is uniformly convex, while the $\ell^1$-norm isn't. Of course, $\mathbb{R}^2$ is reflexive with respect to both norms.
The answer your updated second question is a classical result of M.M. Day:
This is the main result of his paper bearing the statement of the theorem in its title:
M.M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. Volume 47, Number 4 (1941), 313-317, MR0003446.
Since the paper is freely available via the above link, there is little sense in my elaborating on it.