definition of seminorm on real or complex vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of seminorm 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) (\(0 \le \Vert v_1 \Vert\)) \(\land\) (\((v_1 = 0) \implies (0 = \Vert v_1 \Vert)\))
\(\land\)
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: Note
The difference of 'seminorm' from 'norm' is that being \(0 = \Vert v_1 \Vert\) does not imply being \(v_1 = 0\).
