TITLE: Rudin Functional Analysis Chapter 6 Problem 10
QUESTION [1 upvotes]: I am currently struggling to solve this problem:
Suppose 
$(f_i)$ 
is  a  sequence of locally  integrable  functions  in  $\omega$ (an open set in $\mathbb R$") and 
$\lim_i \int_K |f_i(x)|\,dx = 0$ 
for every compact $K$ in $\omega$ 
Prove that then $D^\alpha (f_i) \rightarrow 0$ in $D'(\omega)$ as i goes to infinity, $\forall$ multi-index $\alpha$
I have tried using the distribution definitions and the derivative of a distribution definition, but I am confused with the terminology and I don't see how to actually solve this. 
Thanks in advance

REPLY [1 votes]: You only have to show that $\int D^{\alpha} (f_i) \phi$ approaches $0.$ By definition of derivative this reduces to showing that $\int D^{\alpha} \phi (f_i)$ approaches $0.$ This is true because $D^{\alpha} \phi$ is bounded on its support $K.$
{"set_name": "stack_exchange", "score": 1, "question_id": 2542347}