definition of 'normed vectors space' isometry
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 map.
Target Context
- The reader will have a definition of 'normed vectors space' isometry.
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
\( V_1\): \(\in \{\text{ the } F \text{ normed vectors spaces }\}\), with any norm, \(\Vert \bullet \Vert_1\)
\( V_2\): \(\in \{\text{ the } F \text{ normed vectors spaces }\}\), with any norm, \(\Vert \bullet \Vert_2\)
\(*f\): \(: V_1 \to V_2\), \(\in \{\text{ the maps }\}\)
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Conditions:
\(\forall v \in V_1 (\Vert f (v) \Vert_2 = \Vert v \Vert_1)\)
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2: Note
Often, "isometry" is used requiring bijectivity, but this definition does not require that because otherwise, we would not have any name for the non-bijective case, while we can just use "bijective isometry" for the bijective case.
Note that the definition does not require linearity.
There can be linear isometries and complex-conjugate-linear isometries as well as non-linear isometries.
Conditions does not equal "\(f\) between the induced metric spaces is a 'metric space' isometry", because \(f\) is not required to be linear: \(dist_2 (f (v_1), f (v_2)) = dist_1 (v_1, v_2)\) equals \(\Vert f (v_1) - f (v_2) \Vert_2 = \Vert v_1 - v_2 \Vert_1\), but \(\Vert f (v_1) - f (v_2) \Vert_2 = \Vert f (v_1 - v_2) \Vert_2\) is not guaranteed, especially, \(\Vert v_1 \Vert_1 = \Vert v_1 - 0 \Vert_1 = \Vert f (v_1) - f (0) \Vert_2\), but \(\Vert f (v_1) - f (0) \Vert_2 = \Vert f (v_1) \Vert_2\) is not guaranteed.
