Suppose $1 \leq p<\infty$, given $f \in L^p (\mathbb{R})$, define $f_n (x)=n^{1/p} f(nx)$ for n=1,2,3... Prove $f_n$ converges weakly to zero in $L^p$.
Now I can just know the that $ \|f_n\|_p$=$ \|f\|_p$, but cannot prove for any function $g \in L_q$, where $q$ is the conjugate of $p$, the integral converges to zero.
i.e. how to show $\int f_ng$ converges to zero. I'm thinking maybe I can use step functions or continuous functions since they are dense in Lp, so we can approximate, but not sure.