definition of metric induced by norm on real or complex vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of norm on real or complex vectors space.
- The reader knows a definition of metric.
Target Context
- The reader will have a definition of metric induced by norm on real or complex vectors space.
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:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( \Vert \bullet \Vert\): \(\in \{\text{ the norms on } V\}\)
\(*dist\): \(: V \times V \to \mathbb{R}, (v_1, v_2) \mapsto \Vert v_2 - v_1 \Vert\), \(\in \{\text{ the metrics on } V\}\)
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2: Note
Let us see that \(dist\) is indeed a metric.
1) \(0 \le dist (v_1, v_2) = \Vert v_2 - v_1 \Vert\) with the equality holding if and only if \(v_1 = v_2\).
2) \(dist (v_1, v_2) = \Vert v_2 - v_1 \Vert = \Vert v_1 - v_2 \Vert = dist (v_2, v_1)\).
3) \(dist (v_1, v_3) = \Vert v_3 - v_1 \Vert = \Vert v_3 - v_2 + v_2 - v_1 \Vert \le \Vert v_2 - v_1 \Vert + \Vert v_3 - v_2 \Vert = dist (v_1, v_2) + dist (v_2, v_3)\), by the definition of norm on real or complex vectors space.
