Discussion on question by Anacardium: How to show that $\|\lambda (f)\| < \infty$ for any $f \in C_c (\mathcal G)\ $?

Imported from a comment discussion on https://math.stackexchange.com/questions/5009599/how-to-show-that-lambda-f-infty-for-any-f-in-c-c-mathcal-g