I'm interested in a question regarding the identification of some duals of quasi-Banach spaces.
However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the right direction regarding a specific question.


First I'll get the definitions out of the way: a quasi-Banach space is a complete quasinormed vector space ('quasinormed' being like 'normed' except now the triangle inequality reads $||x+y|| \leq K(||x|| + ||y||)$ for some $K \geq 1$). The dual space $X^\*$ of a quasi-Banach space $X$ is the set of all continuous linear maps from $X$ into the base field (let's take this to be $\mathbb{C}$), and in general $X^\*$ may be trivial. We say that $X^\*$ separates the points of $X$ if for every $x \in X$ there exists a $x^\* \in X^\*$ such that $x^\*(x) \neq 0$.


**If $X^\*$ separates the points of $X$**, we define the Banach envelope $X\_c$ of $X$ to be the completion of $X$ under the norm $$ ||x||\_{X\_c} := \inf\{\epsilon > 0 : x \in \operatorname{co}(B(0,\epsilon))\},$$
where $\operatorname{co}(B(0,\epsilon))$ is the convex hull of the open ball of radius $\epsilon$ centred at $0$.
The fact that $||\cdot||\_{X\_c}$ is a norm on $X$ supposedly follows from the fact that $X^\*$ separates the points of $X$ (but I'm not familiar with the proof of this fact, so don't quote me on it).


In all the literature I've seen, the Banach envelope of a specific space $X$ is identified after having already identified the dual of $X$ in order to verify that $X^\*$ separates the points of $X$. I would like to use an identification of $X\_c$ in order to identify the dual $X^\*$ (as there is a theorem which says that $X^\* = (X\_c)^\*$). To argue in such a way, I need to know in advance that $X^\*$ separates the points of $X$.


This leads to my question(s), which explain(s) why I've taken all this space discussing Banach envelopes:


*Is there a way of showing that the dual ${X^\*}$ of a quasi-Banach space ${X}$ separates the points of ${X}$ without explicitly identifying ${X^\*}$? Furthermore, if we know that $||\cdot||\_{X\_c}$ is a norm on $X$, does this guarantee that ${X^\*}$ separates the points of $X$?*


(sidenote: posting this question has made me realise that there's no tag explicitly dealing with quasi-Banach spaces, so I don't know how much luck I'll have here!)



Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?
I realised that this is actually not such a difficult question, so I'll answer it myself!


(My solution is somewhat inspired by Jochen Wengenroth's comment. Thanks Jochen)


From the definition of $|| \cdot ||\_{X\_c}$, we immediately have $||x||\_{X\_c} \leq ||x||\_X$ for all $x \in X$. So if $||\cdot||\_{X\_c}$ is a norm on $X$, then the completion $X\_c$ of $(X,||\cdot||\_{X\_c})$ is a Banach space, and $X$ embeds continuously into $X\_c$. If $x \in X$, then there exists an $x^\*$ in the dual $(X\_c)^\*$ such that $x^\*(x) \neq 0$ - because $X\_c$ is a Banach space. But since $X$ embeds continuously into $X\_c$, we have that $x^\*$ restricts to an element of $X^\*$, and therefore we've found $x^\* \in X^\*$ such that $x^\*(x) \neq 0$. So $X^\*$ separates points.


This is good enough for the applications I have in mind, but I'm still interested in other characterisations when $X^\*$ separates points, so feel free to contribute any more that you may have. 


