$L^p$ for $0<p<1$ is not a normed space

Question

I have been trying to prove that $L^p$ for $0<p<1$ is no normed space, however, researching quickly lead me to only find that these spaces are not locally convex. I am trying to understand this without the theroy of local convexity. One author pointed out that the triangle inequality for the corresponsing "norm" object is violated. I have been trying to construct an adequate counterexample (along the lines of $1/x^p$ on an appropriate compact space), but unfortunately I could not come up with one, which worked.

Edit: $f(x) = \frac{a}{x^p}$ for $0<a<1$ works.

Answer 1

If $f=\chi_A$ and $g=\chi_B$ where $A$ and $B$ are disjoint sets with the same positive measure then $\|f+g\|_p >\|f\|_p+\|g\|_p$.