Definition:Supremum Metric
Definition
Let $S$ be a set.
Let $M = \struct {A', d'}$ be a metric space.
Let $A$ be the set of all bounded mappings $f: S \to M$.
Let $d: A \times A \to \R$ be the function defined as:
- $\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in S} \map {d'} {\map f x, \map g x}$
where $\sup$ denotes the supremum.
$d$ is known as the supremum metric on $A$.
Special Cases
Bounded Continuous Mappings
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $A$ be the set of all continuous mappings $f: M_1 \to M_2$ which are also bounded.
Let $d: A \times A \to \R$ be the function defined as:
- $\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in A_1} \map {d_2} {\map f x, \map g x}$
where $\sup$ denotes the supremum.
$d$ is known as the supremum metric on $A$.
Bounded Real-Valued Functions
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 $A$.
Bounded Real Sequences
Let $A$ be the set of all bounded real sequences.
Let $d: A \times A \to \R$ be the function defined as:
- $\ds \forall \sequence {x_i}, \sequence {y_i} \in A: \map d {\sequence {x_i}, \sequence {y_i} } := \sup_{n \mathop \in \N} \size {x_n - y_n}$
where $\sup$ denotes the supremum.
$d$ is known as the supremum metric on $A$.
Bounded Real Functions on Interval
Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $A$ be the set of all bounded real 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 $A$.
Continuous Real Functions
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 $A$.
Differentiability Class
Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $r \in \N$ be a natural number.
Let $\mathscr D^r \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$ which are of differentiability class $r$.
Let $d: \mathscr D^r \closedint a b \times \mathscr D^r \closedint a b \to \R$ be the function defined as:
- $\ds \forall f, g \in \mathscr D^r \closedint a b: \map d {f, g} := \sup_{\substack {x \mathop \in \closedint a b \\ i \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x}$
where:
- $f^{\paren i}$ denotes the $i$th derivative of $f$
- $f^{\paren 0}$ denotes $f$
- $\sup$ denotes the supremum.
$d$ is known as the supremum metric on $\mathscr D^r \closedint a b$.
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:
- $\map {\mathscr B} {X, M}$
but this notation is not universal.
Also see
- Results about the supremum metric can be found here.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.17$