definition of norm on real or complex vectors space
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of norm 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
\( V\): \(\in \{\text{ the vectors spaces over } F\}\)
\(*\Vert \bullet \Vert\): \(: V \to \mathbb{R}\) //
Conditions:
\(\forall v_1, v_2 \in V\), \(\forall r \in F\),
(
1) (\(0 \le \Vert v_1 \Vert\)) \(\land\) (\((0 = \Vert v_1 \Vert) \iff (v_1 = 0)\))
\(\land\)
2) \(\Vert r v_1 \Vert = \vert r \vert \Vert v_1 \Vert\)
\(\land\)
3) \(\Vert v_1 + v_2 \Vert \le \Vert v_1 \Vert + \Vert v_2 \Vert\)
)
//
2: Natural Language Description
For any field, \(F \in \{\mathbb{R}, \mathbb{C}\}\), and any vectors space, \(V\), over \(F\), any map, \(\Vert \bullet \Vert: V \to \mathbb{R}\), such that for each vectors, \(v_1, v_2 \in V\), and each scalar, \(r \in F\), 1) \(\Vert v_1 \Vert \ge 0\) with the equality holding if and only if \(v_1 = 0\); 2) \(\Vert r v_1 \Vert = \vert r \vert \Vert v_1 \Vert\); 3) \(\Vert v_1 + v_2 \Vert \le \Vert v_1 \Vert + \Vert v_2 \Vert\)
