Basic Analysis I — Real Analysis · Problem 7.4.11

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

K

be the set of

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

such that

f

is equal to a quadratic polynomial, i.e.

f(x) = a + bx + cx^2

, and such that

|f(x)| \leq 1

for all

x \in [0,1]

, that is

f \in C(0,1)

. Show that

K

is compact.