Let $K$ be a compact [topological space](/page/Topological%20Space), and let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $C(K;\mathbb{F})$ denote the [vector space](/page/Vector%20Space) of continuous functions $f: K \to \mathbb{F}$. Equip $C(K;\mathbb{F})$ with the [uniform norm](/page/Uniform%20Norm) $\|f\|_\infty := \sup_{x \in K} |f(x)|$, with the convention that this supremum is $0$ when $K = \varnothing$. Then $(C(K;\mathbb{F}), \|\cdot\|_\infty)$ is a [Banach space](/page/Banach%20Space).