Uniform metric

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

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

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.