Supremum Metric on Differentiability Class is Metric
Theorem
Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $r \in \N$ be a natural number.
Let $A := \mathscr D^r \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$ which are of differentiability class $r$.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric.
Proof
We have that the supremum metric on $A \times A$ is defined as:
- $\ds \forall f, g \in A: \map d {f, g} := \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x}$
where $f$ and $g$ are continuous functions on $\closedint a b$ which are of differentiability class $r$.
Proof of Metric Space Axiom $(\text M 1)$
\(\ds \map d {f, f}\) | \(=\) | \(\ds \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {f^{\paren i} } x}\) | Definition of $d$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
So Metric Space Axiom $(\text M 1)$ holds for $d$.
$\Box$
Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality
Let $f, g, h \in A$.
Let $c \in \closedint a b$.
\(\ds c\) | \(\in\) | \(\ds \closedint a b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sup_{i \mathop \in \set {0, 1, 2, \ldots, r} } \size {\map {f^{\paren i} } c - \map {h^{\paren i} } c}\) | \(\le\) | \(\ds \sup_{i \mathop \in \set {0, 1, 2, \ldots, r} } \size {\map {f^{\paren i} } c - \map {g^{\paren i} } c} + \sup_{i \mathop \in \set {0, 1, 2, \ldots, r} } \size {\map {g^{\paren i} } c - \map {h^{\paren i} } c}\) | Triangle Inequality for Real Numbers | ||||||||||
\(\ds \) | \(\le\) | \(\ds \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x} + \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {g^{\paren i} } x - \map {h^{\paren i} } x}\) | Definition of Supremum of Real-Valued Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map d {f, g} + \map d {g, h}\) | Definition of $d$ |
Thus $\map d {f, g} + \map d {g, h}$ is an upper bound for:
- $\ds S := \set {\sup_{i \mathop \in \set {0, 1, 2, \ldots, r} } \size {\map {f^{\paren i} } c - \map {g^{\paren i} } c}: c \in \closedint a b}$
So:
- $\map d {f, g} + \map d {g, h} \ge \sup S = \map d {f, h}$
So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d$.
$\Box$
Proof of Metric Space Axiom $(\text M 3)$
\(\ds \map d {f, g}\) | \(=\) | \(\ds \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x}\) | Definition of $d$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {g^{\paren i} } x - \map {f^{\paren i} } x}\) | Definition of Absolute Value | |||||||||||
\(\ds \) | \(=\) | \(\ds \map d {g, f}\) | Definition of $d$ |
So Metric Space Axiom $(\text M 3)$ holds for $d$.
$\Box$
Proof of Metric Space Axiom $(\text M 4)$
As $d$ is the supremum of the absolute value of the image of the pointwise sum of $f$ and $g$:
- $\forall f, g \in A: \map d {f, g} \ge 0$
Suppose $f, g \in A: \map d {f, g} = 0$.
Then:
\(\ds \map d {f, g}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {f^{\paren i} } x}\) | \(=\) | \(\ds 0\) | Definition of $d$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall x \in \closedint a b: \, \) | \(\ds \map f x\) | \(=\) | \(\ds \map g x\) | Definition of Absolute Value | |||||||||
\(\ds \leadsto \ \ \) | \(\ds f\) | \(=\) | \(\ds g\) | Equality of Mappings |
So Metric Space Axiom $(\text M 4)$ holds for $d$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.17$