I have a quick question which is probably supposed to be obvious, but for some reason I just don't see it: How does one re-norm a quasi-Banach space to produce a $p$-Banach space ($0<p\leq 1$) with the same topology?
A quasi-Banach space is, of course, just like a Banach space except the triangle inequality requirement for the norm is relaxed to $\|x+y\|\leq\kappa(\|x\|+\|y\|)$ for some $1\leq\kappa<\infty$, producing a "quasi-norm" (the topology remains the $\varepsilon$-ball topology, which is metrizable even if not always metric, so that we can talk about completions). A special case of quasi-Banach space are "$p$-Banach spaces" ($0<p\leq 1$) which satisfy $\|x+y\|^p\leq\|x\|^p+\|y\|^p$ (and then we can take $\kappa=2^{1/p-1}$).
This paper claims that every quasi-Banach space admits an equivalent $p$-norm for some $0<p\leq 1$. Here is a quotation for context:
The Aoki-Rolewicz’s Theorem states that any quasi-Banach space $\mathbb{X}$ is $p$-convex for some $0<p\leq 1$, i.e., there is a constant $C$ such that
$$\left\|\sum_{j=1}^nf_j\right\|\leq C\left(\sum_{j=1}^n\|f_j\|^p\right)^{1/p},\;\;n\in\mathbb{N},\;\;f_j\in\mathbb{X}.$$
This way, $\mathbb{X}$ becomes a quasi-Banach space under a suitable renorming.
I'm sorry if I'm missing something obvious, but what is that "suitable renorming", exactly?
Thanks!
