Equicontinuity and uniform boundedness for "distributions"

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.

Solution 1:

Denote by $p_i(\phi) = \sup_{x \in [-1,1]} |D^i \phi(x)|$ the usual seminorms on $\mathcal{D}$. Observe that $p_i \leq 2 p_{i+1}$ for all $i\in\mathbb{N}$. By the equicontinuity of the $\Lambda_n$ there is a $\delta>0$ and a neighbourhood $$ U = \{ \phi \in \mathcal{D}; p_{i_1}(\phi) < \delta, ..., p_{i_k}(\phi)<\delta\}$$ of $0$ such that $|\Lambda_n(\phi)| \leq 1$ for all $\phi \in U$ and $n\in \mathbb{N}$. Without loss of generality assume $i_1 < ... < i_k$. Now let $\phi\neq 0$ and $M:=\max\{p_{i_1}(\phi),...,p_{i_k}(\phi)\}$. By the observation above $M>0$ and $M\leq 2^{i_k}p_{i_k}(\phi)$. Then $\frac{\delta}{M}\phi \in U$ and hence

$$ \vert\Lambda_n(\phi)| = \vert\frac{M}{\delta} \Lambda_n(\frac{\delta}{M}\phi)\vert \leq \frac{M}{\delta} \leq \frac{2^{i_k}}{\delta} p_{i_k}(\phi). $$