Positive Multiple of Metric is Metric
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $k \in \R_{>0}$ be a (strictly) positive real number.
Let $d_k: A \times A \to \R$ be the function defined as:
- $\forall \tuple {x, y} \in A: \map {d_k} {x, y} = k \cdot \map d {x, y}$
Then $M_k = \struct {A, d_k}$ is a metric space.
Proof
Metric Space Axiom $(\text M 1)$
\(\ds \map {d_k} {x, x}\) | \(=\) | \(\ds k \cdot \map d {x, x}\) | Definition of $d_k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | as $d$ fulfils Metric Space Axiom $(\text M 1)$ |
So Metric Space Axiom $(\text M 1)$ holds for $d_k$.
$\Box$
Metric Space Axiom $(\text M 2)$: Triangle Inequality
\(\ds \map {d_k} {x, y} + \map {d_k} {y, z}\) | \(=\) | \(\ds k \cdot \map d {x, y} + k \cdot \map d {y, z}\) | Definition of $d_k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds k \paren {\map d {x, y} + \map d {y, z} }\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds k \cdot \map d {x, z}\) | as $d$ fulfils Metric Space Axiom $(\text M 2)$: Triangle Inequality | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_k} {x, z}\) | Definition of $d_k$ |
So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d_k$.
$\Box$
Metric Space Axiom $(\text M 3)$
\(\ds \map {d_k} {x, y}\) | \(=\) | \(\ds k \cdot \map d {x, y}\) | Definition of $d_k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds k \cdot \map d {y, x}\) | as $d$ fulfils Metric Space Axiom $(\text M 3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_k} {y, x}\) | Definition of $d_k$ |
So Metric Space Axiom $(\text M 3)$ holds for $d_k$.
$\Box$
Metric Space Axiom $(\text M 4)$
\(\ds x\) | \(\ne\) | \(\ds y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {x, y}\) | \(>\) | \(\ds 0\) | as $d$ fulfils Metric Space Axiom $(\text M 4)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds k \cdot \map d {x, y}\) | \(>\) | \(\ds 0\) | as $k > 0$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_k} {x, y}\) | \(>\) | \(\ds 0\) | Definition of $d_k$ |
So Metric Space Axiom $(\text M 4)$ holds for $d_k$.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $1$