description/proof of that for Euclidean set, taking for \(2\) points, maximum of absolute differences of components of points is metric
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean set.
- The reader knows a definition of maximum of partially-ordered set.
- The reader knows a definition of metric.
- The reader admits the proposition that for any linearly-ordered ring, any finite number of subsets with any same nonempty finite index set, and the subset as the sum of the subsets with the same index set, the maximum of the subset is equal to or smaller than the sum of the maximums of the subsets and the minimum of the subset is equal to or larger than the sum of the minimums of the subsets.
Target Context
- The reader will have a description and a proof of the proposition that for any Euclidean set, taking for each \(2\) points, the maximum of the absolute differences of the components of the points is a metric.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(d\): \(\in \mathbb{N} \setminus \{0\}\)
\(\mathbb{R}^d\): \(= \text{ the Euclidean set }\)
\(f\): \(: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}, (r_1, r_2) \mapsto Max (\{\vert {r_1}^j - {r_2}^j \vert \vert j \in \{1, ..., d\}\})\)
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Statements:
\(f \in \{\text{ the metrics on } \mathbb{R}^d\}\)
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2: Proof
Whole Strategy: Step 1: see that \(f\) satisfies the conditions to be a metric.
Step 1:
Let us see that \(f\) satisfies the conditions to be a metric.
Let \(r_1, r_2, r_3 \in \mathbb{R}^d\) be any.
1) \(0 \le f (r_1, r_2)\) and \(f (r_1, r_2) = 0\) if and only if \(r_1 = r_2\): \(0 \le Max (\{\vert {r_1}^j - {r_2}^j \vert \vert j \in \{1, ..., d\}\})\); if \(f (r_1, r_2) = 0\), \({r_1}^j - {r_2}^j = 0\) for each \(j\), which implies that \(r_1 = r_2\); if \(r_1 = r_2\), \({r_1}^j - {r_2}^j = 0\) for each \(j\), which implies that \(f (r_1, r_2) = 0\).
2) \(f (r_1, r_2) = f (r_2, r_1)\): \(f (r_1, r_2) = Max (\{\vert {r_1}^j - {r_2}^j \vert \vert j \in \{1, ..., d\}\}) = Max (\{\vert {r_2}^j - {r_1}^j \vert \vert j \in \{1, ..., d\}\}) = f (r_2, r_1)\).
3) \(f (r_1, r_3) \le f (r_1, s_2) + f (r_2, r_3)\): \(f (r_1, r_3) = Max (\{\vert {r_1}^j - {r_3}^j \vert \vert j \in \{1, ..., d\}\}) = Max (\{\vert {r_1}^j - {r_2}^j + {r_2}^j - {r_3}^j \vert \vert j \in \{1, ..., d\}\}) \le Max (\{\vert {r_1}^j - {r_2}^j \vert + \vert {r_2}^j - {r_3}^j \vert \vert j \in \{1, ..., d\}\}) \le Max (\{\vert {r_1}^j - {r_2}^j \vert \vert j \in \{1, ..., d\}\}) + Max (\{\vert {r_2}^j - {r_3}^j \vert \vert j \in \{1, ..., d\}\})\), by the proposition that for any linearly-ordered ring, any finite number of subsets with any same nonempty finite index set, and the subset as the sum of the subsets with the same index set, the maximum of the subset is equal to or smaller than the sum of the maximums of the subsets and the minimum of the subset is equal to or larger than the sum of the minimums of the subsets, \(= f (r_1, s_2) + f (r_2, r_3)\).
