Consider the norm $||u||=|x|+|y|$ if $u=(x,y)$. Prove that if $T:\mathbb{R^2}\longrightarrow{\mathbb{R^2}}$ satisfies $||Tu-Tv||=||u-v||$ then $T$ is of the form $Tu=p+Au$ with $A$ linear.

I've been stuck with this problem for a while now, I know what everything is: a norm, a linear application... And their properties, however I just don't know what to do.

  • 1
    This is known as the Mazur-Ulam Theorem. – Theo Bendit Oct 11 at 12:49
  • 1
    @JoséCarlosSantos I disagree. The dupe target uses the Euclidean norm, whereas this question uses the $1$-norm. – Theo Bendit Oct 11 at 12:57
  • 1
    @TheoBendit You are right. I will rectract my closing vote. – José Carlos Santos Oct 11 at 12:57

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.