$l^p$ for $1\leq p<\infty$ is defined as the vector space on which $$x\mapsto\sqrt[p]{\sum_j{|x_j|^p}}$$ is a norm. For $0<p<1$, this cannot be a norm, but (as Wiki indicates) the function $$\mapsto\sum_j{|x_j|^p}$$ is still a metric, so it's a (complete) topological vector space.
If $1\leq p<\infty$, the dual of $l^p$ is well-known to be $l^q$, where $\frac{1}{p}+\frac{1}{q}=1$. What is the (topological) dual of $l^p$ for $0<p<1$?
In that case, $l^p\subseteq l^1$, so $l^{\infty}\subseteq(l^p)^*$. But I don't know what else is in there.