Euclidean Metric is Metric
Theorem
Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
The Euclidean metric on $\AA$ is a metric.
Proof 1
The Euclidean metric on $\AA$ is a special case of the $p$-product metric.
The result follows from $p$-Product Metric is Metric.
$\blacksquare$
Proof 2
We have that the Euclidean metric on $\AA$ is defined as:
- $\ds \map {d_2} {x, y} = \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.
Proof of Metric Space Axiom $(\text M 1)$
\(\ds \map {d_2} {x, x}\) | \(=\) | \(\ds \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, x_i} }^2}^{\frac 1 2}\) | Definition of $d_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum_{i \mathop = 1}^n 0^2}^{\frac 1 2}\) | as $d_{i'}$ fulfils Metric Space Axiom $(\text M 1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
So Metric Space Axiom $(\text M 1)$ holds for $d_2$.
$\Box$
Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality
Let:
- $(1): \quad z = \tuple {z_1, z_2, \ldots, z_n}$
- $(2): \quad$ all summations be over $i = 1, 2, \ldots, n$
- $(3): \quad \map {d_{i'} } {x_i, y_i} = r_i$
- $(4): \quad \map {d_{i'} } {y_i, z_i} = s_i$.
Thus we need to show that:
- $\ds \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2} + \paren {\sum \paren {\map {d_{i'} } {y_i, z_i} }^2}^{\frac 1 2} \ge \paren {\sum \paren {\map {d_{i'} } {x_i, z_i} }^2}^{\frac 1 2}$
We have:
\(\ds \map {d_2} {x, y} + \map {d_2} {y, z}\) | \(=\) | \(\ds \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2} + \paren {\sum \paren {\map {d_{i'} } {y_i, z_i} }^2}^{\frac 1 2}\) | Definition of $d_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum r_i^2}^{\frac 1 2} + \paren {\sum s_i^2}^{\frac 1 2}\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds \paren {\sum \paren {r_i + s_i}^2}^{\frac 1 2}\) | Minkowski's Inequality for Sums: index $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} + \map {d_{i'} } {y_i, z_i} }^2}^{\frac 1 2}\) | Definition of $r_i$ and $s_i$ | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \paren {\sum \paren {\map {d_{i'} } {x_i, z_i} }^2}^{\frac 1 2}\) | as $d_{i'}$ fulfils Metric Space Axiom $(\text M 2)$: Triangle Inequality | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_2} {x, z}\) | Definition of $d_2$ |
So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d_2$.
$\Box$
Proof of Metric Space Axiom $(\text M 3)$
\(\ds \map {d_2} {x, y}\) | \(=\) | \(\ds \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}\) | Definition of $d_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum \paren {\map {d_{i'} } {y_i, x_i} }^2}^{\frac 1 2}\) | as $d_i$ fulfils Metric Space Axiom $(\text M 3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_2} {y, x}\) | Definition of $d_2$ |
So Metric Space Axiom $(\text M 3)$ holds for $d_2$.
$\Box$
Proof of Metric Space Axiom $(\text M 4)$
\(\ds x\) | \(\ne\) | \(\ds y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists k \in \closedint 1 n: \, \) | \(\ds x_k\) | \(\ne\) | \(\ds y_k\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_k} {x_k, y_k}\) | \(>\) | \(\ds 0\) | as $d_k$ fulfils Metric Space Axiom $(\text M 4)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}\) | \(>\) | \(\ds 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_2} {x, y}\) | \(>\) | \(\ds 0\) | Definition of $d_2$ |
So Metric Space Axiom $(\text M 4)$ holds for $d_2$.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Pythagoras' Theorem
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.7$