A 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: Definition
For any real or complex vectors space, \(V\), and any norm, \(\Vert \bullet \Vert: V \to \mathbb{R}\), the metric, \(dist: V \times V \to \mathbb{R}\), such that \(dist (p_1, p_2) = \Vert p_2 - p_1 \Vert\) for any \(p_1, p_2 \in V\)
2: Note
That \(dist\) is indeed a metric, because 1) \(0 \le dist (p_1, p_2) = \Vert p_2 - p_1 \Vert\) with the equality holding if and only if \(p_1 = p_2\); 2) \(dist (p_1, p_2) = \Vert p_2 - p_1 \Vert = \Vert p_1 - p_2 \Vert = dist (p_2, p_1)\); 3) \(dist (p_1, p_3) = \Vert p_3 - p_1 \Vert = \Vert p_3 - p_2 + p_2 - p_1 \Vert \le \Vert p_2 - p_1 \Vert + \Vert p_3 - p_2 \Vert = dist (p_1, p_2) + dist (p_2, p_3)\), by the definition of norm on real or complex vectors space.
