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

[wayneckm]

Hello all,


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

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

Thanks a lot.


Wayne

 

[micromass]

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