Some properties about $L^p$ with $0<p<1$

Question

We are coming across many Banach spaces $L^p$ with $1\leq p\leq\infty$. But how about $0<p<1$? Can it be normed? How about its metric induced by the norm? And how about its convexity,completeness and reflexiveness? And moreover,can we define functionals on it?

Answer 1

Since the map $t\mapsto t^p$ is sub-additive, one can define on $L^p$ the metric $$\rho(f,g):=\int_X|f(x)-g(x)|^p\mathrm dx.$$ This metric is complete.

However it can be shown that the topological vector space $L^p$ with this metric is not locally convex ($0\lt p\lt 1$). Furthermore, the topological dual of this topological vector space is reduced to the null functional.