The supremum metric $d_\infty(f,g) = \sup_x |f(x) - g(x)|$ is a metric on the bounded real-valued functions on a nonempty set

Let $S$ be a nonempty set. Call a function $f : S \to \mathbb{R}$ bounded when its range $f[S] = \{f(s) : s \in S\}$ is a bounded subset of $\mathbb{R}$ (Lower bound, bounded below, bounded set), and write

$\mathcal{B}(S) := \{\, f : f \text{ is a bounded function } S \to \mathbb{R} \,\}.$

For $f, g \in \mathcal{B}(S)$ put $D(f,g) := \{\, |f(s) - g(s)| : s \in S \,\}$ and

$d_\infty(f,g) := \sup D(f,g).$

This is well defined: $D(f,g)$ is nonempty because $S$ is, and it is bounded above (step 1.1 below), so its least upper bound exists (Complete ordered field (least-upper-bound property)) and is unique (Suprema and infima are unique).

Then $d_\infty$ is a metric on $\mathcal{B}(S)$ (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric), the supremum metric (also called the uniform metric).

The hypotheses ensure that the formula is a finite real-valued metric for every pair in the stated function space. Boundedness of $f$ and $g$ makes $D(f,g)$ bounded above, and nonemptiness of $S$ makes it nonempty. Some unbounded pairs can still have a finite supremum, but allowing all real-valued functions would not give a finite-valued metric: for example, on $S=\mathbb{R}$ the functions $f(s)=s$ and $g(s)=0$ make $D(f,g)$ unbounded above (Conventions: $\sup \emptyset$, unbounded sets, and the extended reals).

• Why the bounded functions and not all functions. For unbounded $f$ the set $D(f,g)$ need not be bounded above and $\sup D(f,g)$ then does not exist, so $d_\infty$ would not be a function into $\mathbb{R}$ at all. Texts that write $d_\infty(f,g) = +\infty$ in that case are working in the extended real line, which is introduced on a later page. The suprema taken here are real numbers, and the extended real line is not used for them (Conventions: $\sup \emptyset$, unbounded sets, and the extended reals, Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here).
• The supremum need not be attained, so $d_\infty(f,g)$ is genuinely a supremum and not a maximum; the companion page carries a witness.
• The name "uniform metric" points at later material. The quantified definition of uniform convergence of functions appears in Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions ↗. The later Agreement of the quantified real-valued definition with the later uniform-metric and uniform-topology formulations ↗ records its agreement with convergence in $d_\infty$; this lemma proves only that $d_\infty$ is a metric on the stated bounded-function space.