definition of equivalence relation on norms on 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
- 3: Note
Starting Context
- The reader knows a definition of norm on real or complex vectors space.
- The reader knows a definition of equivalence relation on set.
Target Context
- The reader will have a definition of equivalence relation on norms on 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:
\(V\): \(\in \{\text{ the real or complex vectors spaces }\}\)
\(S\): \(= \{\text{ the norms on } V\}\)
\(*\sim\): \(\subseteq S \times S\), \(\in \{\text{ the equivalence relations on } S\}\)
//
Conditions: \(\forall s_1, s_2 \in S\)
(
\(s_1 \sim s_2\)
\(\iff\)
\(\exists c_1, c_2 \in \mathbb{R} \text{ such that } 0 \lt c_1, c_2 (\forall v \in V (c_1 s_2 (v) \le s_1 (v) \le c_2 s_2 (v)))\)
)
//
2: Natural Language Description
For any real or complex vectors space, \(V\), and the set of the norms on \(V\), \(S\), the equivalence relation, \(\sim \subseteq S \times S\), such that for each \(s_1, s_2 \in S\), \(s_1 \sim s_2\) if and only if there are some \(c_1, c_2 \in \mathbb{R}\) such that \(0 \lt c_1, c_2\) such that \(\forall v \in V (c_1 s_2 (v) \le s_1 (v) \le c_2 s_2 (v)))\)
3: Note
Let us see that \(\sim\) is indeed an equivalence relation.
For each \(s \in S\), \(s \sim s\), because \(c_1 = c_2 = 1\) will do: \(1 s (v) \le s (v) \le 1 s (v)\).
For each \(s_1, s_2 \in S\) such that \(s_1 \sim s_2\), \(s_2 \sim s_1\), because while there are \(c_1, c_2\) such that \(c_1 s_2 (v) \le s_1 (v) \le c_2 s_2 (v)\), \({c_2}^{-1} s_1 (v) \le s_2 (v) \le {c_1}^{-1} s_1 (v)\).
For each \(s_1, s_2, s_3 \in S\) such that \(s_1 \sim s_2\) and \(s_2 \sim s_3\), \(s_1 \sim s_3\), because while there are \(c_1, c_2\) such that \(c_1 s_2 (v) \le s_1 (v) \le c_2 s_2 (v)\) and \(c'_1, c'_2\) such that \(c'_1 s_3 (v) \le s_2 (v) \le c'_2 s_3 (v)\), \(c_1 c'_1 s_3 (v) \le s_1 (v) \le c_2 c'_2 s_3 (v)\).
