Let Cla, b] be the set of continuous functions f [a,b] R. Given two functions f and g, we can define and cb p(f.g) = |* | f(x) g(x)\dx PM (f. 9) = max ze[a,b] {f(x) - g(x)}. (a) Prove that p and PM are metrics. (b) Use Theorem 5.15 to prove that the topology induced by PM on C[a, b] is finer than topology induced by p. THEOREM 5.15. Let d and d' be metrics on a set X, and let T and T', respectively, be the topologies that they induce. Then T' is finer than T if and only if for each xe X and & > 0, there exists a &> 0 such that Ba (x, 8) C Ba(x, 8). (See Figure 5.8.) Bd (x, e) B (x, 6) FIGURE 5.8: For each x X and > 0, there exists & such that Bar (x, 8) C Ba(x, 8).