Reflexive and strictly convex but not uniform convex

Question

I’m struggling with one question, I can find that authors are writing over uniform convex in Banach spaces a lot but still I haven’t found a good exampel for this:

If space X is reflexive and strictly convex then this is not implying uniform convex.

We can start easily with $\ell^2 $products and $x,y \in R^2$ with p-norm:

Where $||(x,y)||=(|x|^p+|y|^p)^{\frac{1}{p}}$

And we can see that when p>2 so p=3,4,5… Norm is going to far from center of space and that’s why this is not uniform convex but how to write it properly?

Answer 1

Let $\ell_2^p$ denote $\mathbb{C}^2$ with the $p$ norm. Choose a decreasing sequence $(p_k)$ with $p_k\to 1$ and $p_k>1\ \forall k\in\mathbb{N}$. Let $$\displaystyle X = (\bigoplus_k \ell_2^{p_k})_{\ell^2}.$$ That is, $X$ is the space of sequences with the norm $$\|(a_k)\| = \left(\sum_{k\in\mathbb{N}} (|a_{2k}|^{p_k} + |a_{2k+1}|^{p_k})^\frac{2}{p_k}\right)^\frac{1}{2} <\infty .$$ $X$ is reflexive, strictly convex, but not uniformly convex.