Taxicab Metric on Real Vector Space is Metric
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Theorem
The taxicab metric on the real vector space $\R^n$ is a metric.
Proof 1
This is an instance of the taxicab metric on the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
This is proved in Taxicab Metric is Metric.
$\blacksquare$
Proof 2
The taxicab metric on $\R^n$ is:
- $\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \size {x_i - y_i}$
for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
Proof of Metric Space Axiom $(\text M 1)$
\(\ds \map {d_1} {x, x}\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {x_i - x_i}\) | Definition of $d_1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
So Metric Space Axiom $(\text M 1)$ holds for $d_1$.
$\Box$
Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality
\(\ds \map {d_1} {x, y} + \map {d_1} {y, z}\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {x_i - y_i} + \sum_{i \mathop = 1}^n \size {y_i - z_i}\) | Definition of $d_1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \paren {\size {x_i - y_i} + \size {y_i - z_i} }\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds \sum_{i \mathop = 1}^n \size {x_i - z_i}\) | Triangle Inequality for Real Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_1} {x, z}\) | Definition of $d_1$ |
So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d_1$.
$\Box$
Proof of Metric Space Axiom $(\text M 3)$
\(\ds \map {d_1} {x, y}\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {x_i - y_i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {y_i - x_i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_1} {y, x}\) | Definition of $d_1$ |
So Metric Space Axiom $(\text M 3)$ holds for $d_1$.
$\Box$
Proof of Metric Space Axiom $(\text M 4)$
\(\ds x\) | \(\ne\) | \(\ds y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists i \in \set {1, 2, \ldots, n}: \, \) | \(\ds x_i\) | \(\ne\) | \(\ds y_i\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {x_i - y_i}\) | \(>\) | \(\ds 0\) | Definition of Absolute Value | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{i \mathop = 1}^n \size {x_i - y_i}\) | \(>\) | \(\ds 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_1} {x, y}\) | \(>\) | \(\ds 0\) | Definition of $d_1$ |
So Metric Space Axiom $(\text M 4)$ holds for $d_1$.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $2$