definition of Euclidean norm on Euclidean vectors space
Topics
About: normed vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of Euclidean inner product on Euclidean vectors space.
- The reader knows a definition of norm induced by inner product on real or complex vectors space.
Target Context
- The reader will have a definition of Euclidean norm on Euclidean 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:
\( \mathbb{R}^d\): \(= \text{ the Euclidean vectors space with the Euclidean inner product }\)
\(*\Vert \bullet \Vert\): \(= \text{ the norm induced by the Euclidean inner product }\)
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Conditions:
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2: Natural Language Description
For the Euclidean vectors space, \(\mathbb{R}^d\), the norm, \(\Vert \bullet \Vert\), induced by the Euclidean inner product
3: Note
It is indeed a norm, because the norm induced by any inner product on any real or complex vectors space is a norm, by the proposition that any inner product on any real or complex vectors space induces a norm.
The normed Euclidean vectors space, \(\mathbb{R}^d\), does not need to really have the Euclidean inner product: the inner product is used to just in order to define the norm, and the inner product can be forgotten after that if one likes so. In fact, the Euclidean norm can be defined without the Euclidean inner product, but this definition uses the Euclidean inner product in order to skip independently proving that it is indeed a norm.
Although a Euclidean vectors space tends to be implicitly supposed to have the Euclidean norm, it is not necessarily so.
