In fact, I have ever tried to prove in the same way. But I met a problem on one step. More precisely, for $f\in L^1$, define $T_f(\phi):=\int_\Omega f(x)\phi(x)dx$ $\forall \phi\in H^m$. It follows from $H^m\hookrightarrow L^\infty$ that $T_f\in (H^m)^*$ and $\|T_f\|\leq \|f\|_{L^1}$. Moreover, the mappling $T:f\longrightarrow T_f$ is a bijection from $L^1$ onto $T(L^1)$ which is a subset of $(H^m)^*$. We are left to prove that $\|T_f\|=\|f\|_{L^1}$ such that $T$ is a Isometric isomorphism. My question is how to prove that $\|f\|_{L^1}\leq \|T_f\|$? Thank you very much.

@ Math604. Thank you so much. Could you please show a proof by using the imbedding of Sobolev space into $L^\infty$?

@ Jochen Glueck. Thank you so much for your answer.