Sup norm on bounded continuous functions

Example

Let $X$ be a topological space and let $C_b(X)$ be the real vector space of bounded continuous maps $f \colon X \to \mathbb{R}$. For $f \in C_b(X)$, let $\|f\|_\infty = \sup_{x \in X} |f(x)|$, with $\|f\|_\infty = 0$ when $X$ is empty. Then $\|\cdot\|_\infty$ is a norm on $C_b(X)$, the sup norm, and $C_b(X)$ is a Banach space with this norm.

Remarks

The supremum is finite because $f$ is bounded. The metric $\|f - g\|_\infty$ is the uniform metric. Its completeness is completeness of the space of bounded continuous functions. A sequence converges in the sup norm if and only if it converges uniformly. For a compact space $X$, each continuous real function is bounded, so $C_b(X) = C(X)$.

Explanations

L1 human

The sup norm of a function is the height of the highest point of its graph, in absolute value. Two functions are close in this norm when their graphs are close everywhere.