Prove that every continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is of the form $\Lambda\mapsto\Lambda f$ for some distribution $\Lambda$ with compact support.
I am stuck at this problem, which is a slightly modified version of an exercise from Rudin's Functional Analysis (Ex 19, chapter 6). There is a theorem in the book that states the same conclusion for the space of test functions $D(\Omega)$, but I can't modify its proof for the stated problem.
Any hint would be appreciated.
Does Rudin really use the notation $C^\infty_{loc}$? This makes little sense to me; I can't imagine how $C^\infty_{loc}$ could be anything other than $C^\infty$.
First we need to note that taken literally the conclusion makes no sense: By definition a distribution is a (continuous) linear functional on $C^\infty_c$, so there's simply no such thing as $\Lambda f$ for $\Lambda$ a distribution and $f\in C^\infty$.
Say $L$ is a continuous linear functional on $C^\infty$. Saying that $L$ is given by a distribution with compact suppport actually means this: There exists a distribution with compact support $\Lambda$ such that $Lf=\Lambda\phi$ whenever $f\in C^\infty$, $\phi\in C^\infty_c$, and $\phi=f$ on a neighborhood of the support of $\Lambda$.
Hint: The topology is uniform convergence of all derivatives on compact sets. That is to say, the topology induced by the seminorms $$\rho_{K,N}(f)=\sup_{x\in K}\sum_{|\alpha|\le N}|D^\alpha f(x)|$$for compact sets $K$ and positive integers $N$. Any finite family of these seminorms is dominated by one of these seminorms; hence there exist $K$, $N$ and $c$ so that $$|L f|\le c\rho_{K,N}(f).$$
Now define $\Lambda$ to be the restriction of $L$ to $C^\infty_c$. The hint shows that $\Lambda$ is in fact a distribution with compact support, and also that if $\phi$ is a test function with $\phi=f$ in a neighborhood of $K$ then $Lf=L\phi=\Lambda\phi$.
There was a second hint in the original version of this post:
Second hint, edited: For every compact $K$ there exists $\psi\in C^\infty_c$ such that $\psi=1$ on a neighborhood of $K$.
On reflection that's actually irrelevant here; it shows that any distribution with compact support ``is'' a continuous linear functional on $C^\infty$. (If $\Lambda$ is a distibution with compact support define $L:C^\infty\to\mathbb C$ by $Lf=\Lambda(f\psi)$...)