function by where Let (X, d) b a compact metric space and a > 0. We define the following M av m || || Ca (X;R) R VC (XR) ||f||a = ||f|| + sup {\f(z) - 5(!)); 2,4 X, x + 1}. f(y)] x, y 0 [d(x, y)]* ||f|| = sup{\f(x) : x X}. BA (a) Show that the function given by (1) is a norm on the space Ca(X;R). BA (b) Show that if a then C(X; R) C Ca(X; R); (c) Show that the space Ca(X; R) equipped with the norm | |, is a Banach space. I 00,0 (1)