definition of Euclidean metric on Euclidean set
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean norm on Euclidean vectors space.
- The reader knows a definition of metric induced by norm on real or complex vectors space.
Target Context
- The reader will have a definition of Euclidean metric on Euclidean set.
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:
\( \mathbb{R}^d\): \(= \text{ the Euclidean set }\)
\(*dist\): \(= \text{ the metric for } \mathbb{R}^d \text{ induced by the Euclidean norm for the Euclidean vectors space }, \mathbb{R}^d\)
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Conditions:
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2: Note
The Euclidean metric space, \(\mathbb{R}^d\), does not need to really have the Euclidean vectors space structure or the Euclidean norm: the norm is used just in order to define the metric, and the norm and the vectors space structure can be forgotten after that if one likes so. In fact, the metric can be defined without the norm, but this definition uses the norm in order to use the fact that the metric induced by any norm is indeed a metric.
