Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.3

Nonautonomous Linear Systems

Mathematics — Calculus · complete-metric-space-continuous-functions

Problem

Let

I \subset \mathbb{R}

be a closed, bounded interval, let

B \subset \mathbb{R}^n

be a closed ball, and let

C^0(I, B)

denote the set of continuous functions from

I

B

. Define the sup norm

\|f\|_\infty = \sup_{t \in I} \|f(t)\|_{\mathbb{R}^n}.

Show that

C^0(I, B)

is a complete metric space with the distance function

d(f, g) = \|f - g\|_\infty.