Continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is a distribution. 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.    
 A: 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 $Lf=\Lambda(f\psi)$...)
A: I wonder if the point is that the distribution can be chosen to have compact support. This is true if you consider the functional as acting on functions whose support is contained in a fixed bounded set.
