# quick question about renorming quasi-Banach spaces into p-Banach spaces

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) 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" ($$0) which satisfy $$\|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. Here is a quotation for context:

The Aoki-Rolewicz’s Theorem states that any quasi-Banach space $$\mathbb{X}$$ is $$p$$-convex for some $$0, i.e., there is a constant $$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!

$$\|x\|^\prime = \inf\Big\{ \big(\sum_{i=1}^n \|x_i\|^p\big)^{1/p}\colon \sum_{i=1}^n x_i = x, x_i\in X, n\in \mathbb N \Big\}\qquad (x\in X)$$
is the standard $$p$$-convex renorming. The hardish part is to find a suitable $$p$$. You will find more details in Kalton & Peck's An F-space sampler.