definition of Lipschitz map from subset of normed vectors space into normed vectors space
Topics
About: vectors space
The table of contents of this article
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.
Target Context
- The reader will have a definition of Lipschitz map from subset of normed vectors space into normed 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:
\( V_1\): \(\in \{\text{ the normed vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the normed vectors spaces }\}\)
\( S\): \(\subseteq V_1\)
\(*f\): \(: S \to V_2\)
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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: \(\exists L \in \mathbb{R} \text{ such that } 0 \le L (\forall s_1, s_2 \in S (\Vert f (s_2) - f (s_1) \Vert \le \Vert s_2 - s_1 \Vert))\)
