Rudin Functional Analysis Chapter 6 Problem 10

Question

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

Answer 1

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 $K.$