757: Equivalence Relation on Norms on Vectors Space

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definition of equivalence relation on norms on vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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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.

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Target Context

• The reader will have a definition of equivalence relation on norms on vectors space.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \{\text{ the real or complex vectors spaces }\}$
$S$: $=\{\text{ the norms on }V\}$
$\ast \sim$: $\subseteq S\times S$, $\in \{\text{ the equivalence relations on }S\}$
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Conditions: $\mathrm{\forall }{s}_1,s_2\in S$
(
$s_1\sim s_2$
$⟺$
$\mathrm{\exists }{c}_1,c_2\in \mathbb{R}\text{ such that }0<c_1,c_2(\mathrm{\forall }v\in V(c_1{s}_2(v)\le s_1(v)\le c_2{s}_2(v)))$
)
//

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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 $0<c_1,c_2$ such that $\mathrm{\forall }v\in V(c_1{s}_2(v)\le s_1(v)\le c_2{s}_2(v)))$

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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: $1s(v)\le s(v)\le 1s(v)$.

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 $c_1^{\prime },c_2^{\prime }$ such that $c_1^{\prime }{s}_3(v)\le s_2(v)\le c_2^{\prime }{s}_3(v)$, $c_1{c}_1^{\prime }{s}_3(v)\le s_1(v)\le c_2{c}_2^{\prime }{s}_3(v)$.

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References

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