definition of norm induced by inner product on real or complex 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 inner product on real or complex vectors space.
- The reader knows a definition of norm on real or complex vectors space.
Target Context
- The reader will have a definition of norm induced by inner product 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: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V\): \(\in \{\text{ the vectors fields over } F\}\) with any inner product, \(\langle \bullet, \bullet \rangle\)
\(*\Vert \bullet \Vert\): \(: V \to \mathbb{R}, v \mapsto \sqrt{\langle v, v \rangle}\)
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Conditions:
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2: Natural Language Description
For the field, \(F \in \{\mathbb{R}, \mathbb{C}\}\), any vectors space, \(V\), over \(F\), with any inner product, \(\langle \bullet, \bullet \rangle\), the norm, \(\Vert \bullet \Vert: V \to \mathbb{R}, v \mapsto \sqrt{\langle v, v \rangle}\)
3: Note
It is indeed a norm, by the proposition that any inner product on any real or complex vectors space induces a norm.
