Let $P=\sum_{|\alpha|\leq k} P^{\alpha}D^{\alpha}$ be a linear differential operator of degree $k$ acting on the smooth sections, $C^{\infty}(M^m, E),$ of a smooth vector bundle over $M.$ For $M=R^m$ and $|P^{\alpha}|\leq C \,\,\forall \, |\alpha|\leq k $, one introduces the space $$ H^k(R^m)=\{ s \in L^2(R^m): {\left\lVert s\right\rVert}^2_k := \int_{R^m} (1+|\eta |^{2k})\,\, |\hat{s}(\eta)|^2 \,d\eta \,\}.$$ I need to show that $P$ extends by continuity to the Sobolev space $H^k (R^m)$, such that $\bar P \in L(H^k(R^m),\, H^0 (R^m)\,).$
To approximate in $L^2 (R^m)$ by smooth sections, lets choose functions $f,g \in C^{\infty} (R^m, [0,1]) $ such that $$f(x)=1, |x|\leq 1, f(x)=0, |x|\geq 2$$ and $$g(x)=0, |x|\leq 1, \int_{|x|\leq 1} g(x)\,dx=1.$$ For $n\in N,$ we define $f_n(x):=f(\frac{x}{n}),\,\, g_n:=n^m g(nx) \implies \int_{|x|\leq 1} g_n(x)\,dx=1.$ Then we can approximate $$ \hat{s}_n(\eta)=\int_{|x|\leq 1} g_n(\eta -\nu )(f_n \hat{s})(\nu)\,d\nu \in C^{\infty}(R^m). $$
Can somebody give me a suggestion how to go further or provide a solution proposal ? Many thanks.