Isometry Problem: Does Convergence in Norm Imply Convergence Under Map?

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In summary, an isometry is a transformation that preserves distances between points. The Isometry Problem is a mathematical problem that asks whether convergence in norm implies convergence under a map. Its significance lies in its applications in various fields and in understanding the concept of convergence. Convergence in norm implies convergence under a map, but the reverse is not always true. While there are some known solutions to the Isometry Problem in specific cases, a general solution has not yet been found.
  • #1
wayneckm
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Hello all,


My question is if [tex] f : X \mapsto Y [/tex] is an isometry which preserves norm, i.e. [tex] \left\| f(x) \right\| _{Y} = \left\| x \right\|_{X} [/tex], does this imply [tex] \left\| f(x_2) - f(x_1) \right\| _{Y} = \left\| x_2 - x_1 \right\|_{X} [/tex]?

Or, essentially is it sufficient to gurantee convergence in [tex] X [/tex]-norm implies convergence in [tex] Y[/tex]-norm under the map [tex] f [/tex]?

Thanks a lot.


Wayne
 
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  • #2
If your f is linear, then this will be true. Otherwise, it may fail...
 

1. What is an isometry?

An isometry is a transformation that preserves distances between points. In other words, when an object is transformed by an isometry, the distance between any two points on the object remains the same.

2. What is the Isometry Problem?

The Isometry Problem is a mathematical problem that asks whether convergence in norm implies convergence under a map. In other words, if a sequence of points converges under a certain distance measure, will it also converge under a different distance measure?

3. What is the significance of the Isometry Problem?

The Isometry Problem is significant because it helps us understand the concept of convergence and how it relates to different distance measures. It also has applications in fields such as physics, engineering, and computer science.

4. What is the relationship between convergence in norm and convergence under a map?

Convergence in norm implies convergence under a map, but the reverse is not always true. This means that if a sequence of points converges under a certain distance measure, it will also converge under a different distance measure. However, if a sequence converges under a map, it may not necessarily converge in norm.

5. Are there any known solutions to the Isometry Problem?

There are some known solutions to the Isometry Problem in specific cases, but a general solution has not yet been found. This is an ongoing area of research in mathematics and there is still much to be discovered about this problem.

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