# 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 $2.2.17$