Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm?
Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a space that does not have a strictly convex norm?
Notes:
- every separable space has an equivalent strictly convex norm;
- the classical examples of spaces without a strictly convex norm include $\ell_\infty / c_0$ (see also this post) and $\ell_\infty(\Gamma)$ for any uncountable set $\Gamma$.
This problem is attributed to Antonio Avilés; see Question 7.7 in:
- J. Garbulińska and W. Kubiś, Remarks on Gurarii spaces, Extracta Math. 26 (2011), 235–269.
however I learnt that only having already asked it myself.
