446: Metric Induced by Norm on Real or Complex Vectors Space

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A definition of metric induced by norm on real or complex vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of norm on real or complex vectors space.
• The reader knows a definition of metric.

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Target Context

• The reader will have a definition of metric induced by norm on real or complex vectors space.

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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.

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Main Body

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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 $p_1,p_2\in V$

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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.

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References

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