# Prove that if $||Tu-Tv||=||u-v||$ then $T$ is of the form $Tu=p+Au$ with $A$ linear.

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.

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