Basic Analysis I — Real Analysis · Problem 7.1.12

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · metric-spaces

Problem

Let

C^1\bigl([a,b],\mathbb{R}\bigr)

be the set of once continuously differentiable functions on

[a,b]

. Define

d(f,g) \coloneq \|f - g\|_{[a,b]} + \|f' - g'\|_{[a,b]},

where

\|\cdot\|_{[a,b]}

is the uniform norm. Prove that

d

is a metric.