Definition:Supremum Metric/Bounded Real-Valued Functions
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Definition
Let $S$ be a set.
Let $A$ be the set of all bounded real-valued functions $f: S \to \R$.
Let $d: A \times A \to \R$ be the function defined as:
- $\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in S} \size {\map f x - \map g x}$
where $\sup$ denotes the supremum.
$d$ is known as the supremum metric on $A$.
Also known as
This metric is also known as the sup metric or the uniform metric.
Also see
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.17$