Put $K = [-1,1]$; define $\mathcal{D}_K$ as in section 1.46 with
($\mathbb{R}$ in place of $\mathbb{R}^n$). Suppose $\left\{ f_n
> \right\}$ is a sequence of Lebesgue integrable functions such that$$ \Lambda \phi = \lim_{n,\infty} \int_{-1}^1 f_n(t)\phi(t)dt $$ exists for every
$\phi \in \mathcal{D}_K$. Show that $\Lambda$ is a continuous linear
functional on $\mathcal{D}_K$. Show that there's a positive integer
$p$ and $M < \infty$ such that$$ \left| \int_{-1}^1 f_n(t)\phi(t) dt \right| \leq M \lVert D^p \phi \rVert_\infty $$
Just a note $\mathcal{D}_K$ is the set of all functions in $C^\infty(K)$ whose supports is in $K$.
I really don't have a clue for the existance of $M$ and $p$. For the first part instead I defined
$$
\Lambda_n \phi = \int_{-1}^1 f_n(t)\phi(t)dt
$$
And from here I thought I could use the following theorem, which follows more or less from the Banach-Steinhaus theorem
Theorem 2.8 If $\left\{ \Lambda_n \right\}$ is a sequence of continuous linear mappings from a F-space $X$ into a topological
vector space $Y$, and if $$ \Lambda x = \lim_{n,\infty} \Lambda_n x $$
exists for every $x$ in $X$, then $\Lambda$ is continuous
Since each $\Lambda_n : \mathcal{D}_K \to \mathbb{R}$, and $\mathcal{D}_K$ is a Frechet space, and hence an $F-space$ the only bit to be proven in order to apply theorem 2.8 I need to prove the continuity I'd use
Theorem 1.18
Let $\Lambda$ a linear functionals on a topological vector space $X$.
Assume $\Lambda x \neq 0$ for some $x \in X$. Then each of the
following four properties implies the other threea) $\Lambda$ is continuous
b) The null space $\mathcal{N}(\Lambda)$ is closed
c) $\mathcal{N}(\Lambda)$ is not dense in $X$
d) $\Lambda$ is bounded in some neighborhood $V$ of $0$.
Here is where I start to get confused… If I pick any neighborhood of $0$, call such neighborhood $V$ since $\Lambda_n$ exist for every $\phi$ in $\mathcal{D}_K$ the abs value of the integral is bounded on $V$.
This implies that d) of theorem 1.18 is true, hence $\Lambda_n$ is continuous and by theorem 2.8 $\Lambda$ is continuous.
Is this argument correct?
As said instead for the second part I have no clue at all and any hint would be highly appreciated.
Update
I'm probably missing, among the things, that $M$ and $p$ are indepedent from $n$ but they might change with $\phi$.
Update 2
I've just found this solution, the only bit I'm not entirely convinced is the family of seminorms used, they look different from the one defined in section 1.46 of the book. Al the rest seems to make sense, but I'd still like a confirmation.
Best Answer
From the proof of Theorem 2.8 you get that the family $\{\Lambda_{n}\}_{n}$ is equi-continuous. Hence, given $\varepsilon=1$ there exists an open neighborhood $W$ of the origin such that $\Lambda_{n}(W)\subseteq(-1,1)$ for all $n$. But since the topology of $\mathcal{D}_{K}$ is generated by the norms $\Vert\cdot\Vert_{p}$ you can find $r>0$ and $p\in\mathbb{N}_{0}$ such that $B_{p}(0,r)\subseteq W$, and so $\Lambda_{n}(B_{p}(0,r))\subseteq(-1,1)$ for all $n$, that is, $$ |\Lambda_{n}(\phi)|<1 $$ for all $\phi\in\mathcal{D}_{K}$ with $\Vert\phi\Vert_{p}<r$. If $\phi \in\mathcal{D}_{K}$ and $\phi\neq0$, then $\Vert\phi\Vert_{p}\neq0$ and $$ \Bigl\Vert r\frac{\phi}{2\Vert\phi\Vert_{p}}\Bigr\Vert_{p}<r. $$ By the linearity of $\Lambda_{n}$ it follows that $|\Lambda_{n}(\phi )|\leq2r^{-1}\Vert\phi\Vert_{p}$. Now, $$ \Vert\phi\Vert_{p}=\max\{|D^{n}\phi(x)|:\,x\in K,\,n=0,\ldots,p\}. $$ Using Taylor's formula with center at $-1$ and integral remainder for $\phi$ and all its derivatives of order less than $p$ you get \begin{align*} D^{n}\phi(x) & =D^{n}\phi(-1)+\sum_{k=1}^{p-n-1}\frac{1}{k!}D^{n+k}% \phi(-1)(x+1)^{k}+\frac{1}{(p-n)!}\int_{-1}^{x}D^{p}\phi(t)(x-t)^{p-n}dt\\ & =0+0+\frac{1}{(p-n)!}\int_{-1}^{x}D^{p}\phi(t)(x-t)^{p-n}dt \end{align*} and so $$ |D^{n}\phi(x)|\leq\frac{\Vert D^{p}\phi\Vert_{\infty}}{(p-n)!}\int_{-1}% ^{1}|x-t|^{p-n}dt $$ for all $x\in\lbrack-1,1]$ and all $n=0,\ldots,p-1$. Hence, $$ \Vert\phi\Vert_{p}\leq C\Vert D^{p}\phi\Vert_{\infty}% $$ It follows that$$|\Lambda_{n}(\phi )|\leq2r^{-1}\Vert\phi\Vert_{p}\le 2r^{-1}C\Vert D^{p}\phi\Vert_{\infty}$$