Basic Analysis I — Real Analysis · Problem 7.5.11

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · continuous-functions

Problem

Let

(X, d)

be a compact metric space, let

C(X,\mathbb{R})

be the set of real-valued continuous functions. Define

d(f,g) \coloneq \|f - g\|_X = \sup_{x \in X} |f(x) - g(x)|.

a) Show that

d

makes

C(X,\mathbb{R})

into a metric space. b) Show that for every

x \in X

, the evaluation function

E_x : C(X,\mathbb{R}) \to \mathbb{R}

defined by

E_x(f) \coloneq f(x)

is a continuous function.