If $f_n$ converges to $f$ in $L^p$ then does $f_n^2$ converge to $f^2$?

Question

I want to say no, but what is a counterexample? More generally, does $L^p$ convergence of a sequence of functions imply any sort of convergence (i.e. in some different $L^{p'}$ space) of a function of that sequence of functions?

Answer 1

Yes, it does, just write $$ \|f_n^2-f^2\|_{L^{p/2}} = \|(f_n-f)(f_n+f)\|_{L^{p/2}} $$ and so by Hölder's inequality and the triangle inequality $$ \|f_n^2-f^2\|_{L^{p/2}} ≤ \|f_n-f\|_{L^p}\left(\|f_n\|_{L^p}+\|f\|_{L^p}\right). $$ Since $f_n\to f$ in $L^p$, the sequence $(\|f_n\|_{L^p})_{n\in\Bbb N}$ is bounded and so one deduces that $f_n^2\to f^2$ in $L^{p/2}$.