| 1. | This defines an isometry onto a dense subspace, as required.
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| 2. | The corresponding map is an isometry but in general not onto.
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| 3. | Clearly, every isometry between metric spaces is a topological embedding.
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| 4. | The inverse of a global isometry is also a global isometry.
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| 5. | The inverse of a global isometry is also a global isometry.
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| 6. | Note that ?-isometries are not assumed to be continuous.
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| 7. | It was spurred by the 1987 monograph of quasi-isometry.
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| 8. | Accordingly, analysis of isometry groups is analysis of possible symmetries.
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| 9. | Many physical symmetries are isometries and are specified by symmetry groups.
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| 10. | Thus all sets of Kraus operators are related by partial isometries.
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