Why does the triangle inequality for space $L^p$ with $0

It is said that triangle inequality for the space $L^p(\mathbb{R})$ space doesn't hold if $0<p<1$.
Does anyone know an example for this?

Also, what we can say, for example, about the quantity like $\| f \| \colon =\int_{\mathbb{R}} \sqrt{|f|} dx$?

I think in the space $\{ f ; \|f\| < \infty \}$, triangle inequality $\|f+g\| \le \|f\| + \|g\|$ is valid.