Approximation of norms on Banach spaces (Q1732311)

Recall that, given a Banach space \((X,\|\cdot\|)\) and a property of norms \textbf{P}, we say that \(\|\cdot\|\) can be approximated by norms having \textbf{P} if, for every $\varepsilon > 0$, there exists a norm $|\cdot|$ on $X$ having \textbf{P} such that $\|x\|\leq |x|\leq (1+\varepsilon)\|x\|$ for every $x\in X$. \par The authors find a condition for a Banach space $X$ which guarantees that $(*)$ every equivalent norm on X may be approximated by both $C^\infty$ smooth norms and polyhedral norms. The most important special case of the results of the authors is the following (see Corollary 3.8). Let $X$ have a shrinking symmetric Markushevich basis and let $\theta(f)<\infty$ for all $f\in X^*$. Then $(*)$ holds for $X$. \par We do not give here the definition of $\theta:X^*\to [0,\infty]$ as it is quite technical, we just note that it is quite difficult to verify for concrete spaces. However, the authors provide a nice equivalent condition to ``$\theta(f)<\infty$ for all $f\in X^*$'' in Theorem 4.3 and, using this condition, they prove that several classes of spaces have the property $(*)$. Those include preduals of discrete Lorentz spaces $d(w,1,\Gamma)$, certain symmetric Nakano spaces, and Orlicz spaces. \par Finally, in the last section, the authors find for an arbitrary ordinal number $\alpha$ a scattered compact space $K$ with Cantor-Bendixson rank at least $\alpha$ such that $C(K)$ satisfies $(*)$.