Is the product of $L^1$ functions $L^1$?

Question

Is the product of two $L^1$ functions always $L^1$, and if it's not the case do you have a counter example ? Thank you By $L^1$ i mean functions that are integrable in the sens of Lebesgue.

Answer 1

Let $f(x)=\frac 1 {\sqrt x}$ for $0<x<1$ and $0$ for all other $x$. Then $f\in L^{1}$ but $f^{2} \notin L^{1}$.