Basic Analysis I — Real Analysis · Problem 7.4.8

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · completeness-and-compactness

Problem

(Challenging) Let

C\bigl([0,1],\mathbb{R}\bigr)

be the metric space of Example 7.1.8. Let

0

denote the zero function. Then show that the closed ball

C(0,1)

is not compact (even though it is closed and bounded). Hints: Construct a sequence of distinct continuous functions

\{f_n\}_{n=1}^\infty

such that

d(f_n, 0) = 1

and

d(f_n, f_k) = 1

for all

n \neq k

. Show that the set

\{f_n : n \in \mathbb{N}\} \subset C(0,1)

is closed but not compact. See chapter 6 for inspiration.