Continuity in different Lp spaces

Question

Consider the function $F:C[0,1]\to C[0,1]$, $f\mapsto f^2$. I am trying to figure out whether this is continuous when the starting space uses the $L_p$ norm and the target space uses the $L_q$ norm for $p,q\in\{0,1,\infty\}$. In most cases I have already been able to work it out but I'm quite stuck when $p=2$ and $q=1$. What I need in order for $F$ to be continuous is to show that if $$\lim_{n\to\infty}\int_0^1|f_n-g|^2dx=0,$$ then $$\lim_{n\to\infty}\int_0^1|f_n^2-g^2|dx=0.$$ I don't see why this should be true, but I have not been able to find any counterexample. I've fiddled around with the Cauchy-Schwarz inequality but to no avail. The problem with Cauchy-Schwarz occurs if $\lim_{n\to\infty}\int_0^1|f_n+g|dx=\infty$ since in this case I can't get any decent bound. Any ideas?

Answer 1

$$\int |f_n-g||f_n+g|\leq \sqrt {\int |f_n-g|^{2}} \sqrt {\int |f_n+g|^{2}}.$$ By triangle inequality for the $L^{2}$-norm we get $$\sqrt {\int |f_n+g|^{2}}\leq \sqrt {\int |f_n|^{2}}+\sqrt {\int |g|^{2}}.$$ Also $$\sqrt {\int |f_n|^{2}} \leq \sqrt {\int |f_n-g|^{2}}+\sqrt {\int |g|^{2}}<1+\sqrt {\int |g|^{2}}$$ if $n$ is large enough.