Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{H^k(\mathbb{R}^m)}<M$

Can we say that

$$|\sum\limits_{|\alpha| = k}\sum\limits_{|\beta| = k}\int_{\mathbb{R}^m}D^{\alpha} f D^{\beta}\phi| <N$$

for some $N \in \mathbb{R}, N>0$