Exercise. (Rudin, Functional Analysis, chapter 2, pag. 53). Let us consider the space $ \mathcal D :=\{f \in C^{\infty}(\mathbb R), \, \text{supp}f\subseteq [-1,1] \} $ with the topology induced by the usual topology of $C^{\infty}(\Omega)$. Consider the linear functionals $ \mathcal D \ni \phi \mapsto \Lambda_n \phi :=\int_{[-1,1]}f_n(t)\phi(t)dt \in \mathbb R $ where $\{f_n\}$ is a sequence of Lebesgue integrable functions s.t. $\lim_n \Lambda_n\phi$ esixts for every $\phi \in \mathcal D$.
Using the facts that each $\Lambda_n$ is continuous and, moreover, that $\{\Lambda_n\}_{n \in \mathbb N}$ is equicontinuous I would like to prove that :
There exist two numbers $p \in \mathbb N$, $M \in \mathbb R^+$ s.t. $ \left\vert \int_{[-1,1]}f_n(t)\phi(t)dt \right\vert \le M \Vert D^p\phi \Vert_{\infty} $ for every $n$, for every $\phi \in \mathcal D$.
I think that this is a simple matter of uniform boundedness: we kwow that equicontinuity implies uniform boundedness so we can say $ \forall E \subset \mathcal D \text{ bounded, }\, \exists M > 0 \text{ s.t. } \Lambda_nE \subset [-M,M], \quad \forall n \in \mathbb N. $ I think that this fact is all we need to solve Rudin'exercise, but I do not kwow how to identify bounded sets in $\mathcal D$...
Thanks in advance.