Definition:Supremum Metric/Continuous Real Functions
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Definition
Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $A$ be the set of all continuous functions $f: \closedint a b \to \R$.
Let $d: A \times A \to \R$ be the function defined as:
- $\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in \closedint a b} \size {\map f x - \map g x}$
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.
The metric space $\struct {A, d}$ is denoted in some sources as:
- $\mathscr C \closedint a b$
but this notation is not universal.
Examples
Closure of $\map {B_1} 0$ on $\closedint 0 1$
Let $\closedint 0 1 \subseteq \R$ be the closed unit interval.
Let $\mathscr C \closedint 0 1$ be the supremum space of continuous functions $f: \closedint 0 1 \to \R$.
Then:
- $\map \cl {\map {B_1} \bszero} = \set {f \in \mathscr C \closedint 0 1: \map {d_\infty} {f, \bszero} \le 1}$
where:
- $\map {B_1} \bszero$ denotes the open $1$-ball of $\bszero$
- $\d_\infty$ denotes the Chebyshev distance
- $\bszero$ denotes the constant function $f_0$.
Also see
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.8$