A question on support of a $C^{\infty}$ function.

Question

We have to show that the $\text{supp}(fT)\subseteq \text{supp} (T)$ where $f$ is a $C^{\infty}$function. What i want to know is if we can say this $\langle fT, \phi\rangle = \langle T, f\phi\rangle $? Really appreciate the help.

Answer 1

Yes, by definition $\langle fT, \phi\rangle = \langle T, f\phi\rangle$.