definition of complex Euclidean norm on complex Euclidean vectors space
Topics
About: normed vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of complex Euclidean vectors space.
- The reader knows a definition of complex Euclidean inner product on complex 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 complex Euclidean norm on complex 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:
\( \mathbb{C}^d\): \(= \text{ the complex Euclidean vectors space }\)
\(*\Vert \bullet \Vert\): \(= \text{ the norm for } \mathbb{C}^d \text{ induced by the complex Euclidean inner product }\)
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Conditions:
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2: 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 complex Euclidean vectors space, \(\mathbb{C}^d\), does not need to really have the complex 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 complex Euclidean norm can be defined without the complex Euclidean inner product, but this definition uses the complex Euclidean inner product in order to skip independently proving that it is indeed a norm.
Although a complex Euclidean vectors space tends to be implicitly supposed to have the complex Euclidean norm, it is not necessarily so.
