Lebl-Basic-Analysis-I · Problem 7.1.12

Metric Spaces

Mathematics · metric-spaces-definition

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

where

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

is the uniform norm. Prove that

d

is a metric.