Uniform metric on bounded functions

Example

Let $S$ be a nonempty set. Let $B(S)$ be the set of bounded functions $f \colon S \to \mathbb{R}$, that is, the functions for which there is a real $M$ with $|f(s)| \le M$ for all $s \in S$. For $f, g \in B(S)$, let $$d_\infty(f, g) = \sup_{s \in S} |f(s) - g(s)|.$$ Then $d_\infty$ is a metric on $B(S)$, the uniform metric.

Remarks

The supremum is finite, because $|f(s) - g(s)| \le |f(s)| + |g(s)|$ and both functions are bounded. A sequence $(f_k)$ converges to $f$ in this metric if and only if $f_k \to f$ uniformly on $S$. For $S = \{1, \ldots, n\}$, it is the sup metric on R^n. It is $d_\infty(f, g) = \|f - g\|_\infty$ for the sup norm.

Explanations

L1 human

Two functions are close in the uniform metric when their graphs are close at every point at the same time. For example, on $[0, 1]$ the distance between $f(x) = x$ and $g(x) = x^2$ is $1/4$, the largest gap, at $x = 1/2$.