Uniform metric - Knowlpedia

The uniform metric on a set of bounded real-valued functions on $X$ is defined by

d_\infty(f,g)=\sup_{x\in X} |f(x)-g(x)|.

Equivalently, $d_\infty(f,g)=\|f-g\|_\infty$ where $\|\cdot\|_\infty$ is the supremum norm.

Remarks

This is a metric on the space of bounded functions, and convergence with respect to $d_\infty$ is exactly uniform convergence. Many compactness and approximation results for functions are phrased in terms of this metric on continuous functions.

Examples

On $[0,1]$ , with $f(x)=x$ and $g(x)=0$ , one has $d_\infty(f,g)=1$ .
On $[0,1]$ , for $f_n(x)=x/n$ , $d_\infty(f_n,0)=1/n\to 0$ , so $f_n\to 0$ uniformly.