In spaces $L^p$ for $p \in (0,1)$ the "norm", defined as for $1 \leq p <\infty$, does not satisfies triangle inequality, however satisfy reversed Hoelder and Minkowski inequalities. But $\|\cdot \|^p$ satisfies triangle inequality, condition $\|x\|^p=0$ iff $x=0$, and $\|\lambda x\|^p=|\lambda|^p \|x\|^p$ for scalar $\lambda$ and vectors $x$. Hence, in that case, $L^p$ is F-space (in general not locally convex).
Assume now that $p<0$. If we would define "norm" $\| \cdot\|$ in the standard way (such as for $1 \leq p < \infty$) then it hold reversed Hoelder inequality and reversed Minkowski inequality $\|f+g\|\geq \|f\|+\|g\|$ for positive $f,g$. (I don't know is it true for all $f,g $.)
Is it possible in some natural way to introduce in $L^p$, with negative $p$, norm, metric, topology etc.