1072: Lipschitz Map from Subset of Normed Vectors Space into Normed Vectors Space

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definition of Lipschitz map from subset of normed vectors space into normed 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: Note

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Starting Context

• The reader knows a definition of normed vectors space.
• The reader knows a definition of metric induced by norm on real or complex vectors space.
• The reader knows a definition of metric subspace.
• The reader knows a definition of Lipschitz map between metric spaces.

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Target Context

• The reader will have a definition of Lipschitz map from subset of normed vectors space into normed 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:
$V_1$: $\in \{\text{ the normed vectors spaces }\}$
$V_2$: $\in \{\text{ the normed vectors spaces }\}$
$S$: $\subseteq V_1$
$\ast f$: $:S\to V_2$
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Conditions:
$f\in \{\text{ the Lipschitz maps from }S\text{ into }{V}_2\text{ where }S\text{ is regarded to be a subspace of }{V}_1\text{ regarded to be the metric space induced by the norm and }{V}_2\text{ is regarded to be the metric space induced by the norm }\}$
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2: Note

That condition equals this: $\mathrm{\exists }L\in \mathbb{R}\text{ such that }0\le L(\mathrm{\forall }{s}_1,s_2\in S(\Vert f(s_2)-f(s_1)\Vert \le \Vert s_2-s_1\Vert ))$

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References

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