Exercise 36, § 1

(a) Let X be a locally compact topological space. Prove that the space $\mathscr{C}_0(X)$ possesses the metric approximation property (adapt the proof of Prop. 13 of III, p. 21).

(b) Let X be a locally compact topological space endowed with a positive measure and $p\in [1,+\infty ]$. Prove that the space $L^p(X)$ possesses the metric approximation property (same method).