Let $L^p$ be a topological vector space of measurable functions $f$ such that $\int |f|^p\mathrm{d}\mu<\infty$, and define $$||f||_p := \left( \int |f|^p\mathrm{d}\mu\right)^{1/p}.$$ Is this a quasi-norm? That is, does there exist $K>0$ such that for $f, g\in L^p$ we have $$||f+g||_p \leq K(||f||_p+||g||_p)?$$
Context: By setting $$[f]_p = \left( \sup\limits_{\alpha>0} \alpha^p\mu(\{x : |f(x)|>\alpha\})\right)^{1/p}$$ we define $f$ to be weak $L^p$ if $[f]_p<\infty$. Then $[\cdot]_p$ is a quasi-norm for $0<p<\infty$. It would be nice if the same would hold for $L^p$.
It follows from the inequality $(a+b)^p\leq a^p+b^p$ and the convexity of $x\mapsto x^{1/p}$ for $0<p<1$.