If
$\|f(x)-f(y)\|\geqslant \frac 1{2} \|x-y\|$ for any $x, y \in W$
then
$f$ is injective in $W$
How to prove this? If that inequality is right is it mean that the images are equal or not?
If
$\|f(x)-f(y)\|\geqslant \frac 1{2} \|x-y\|$ for any $x, y \in W$
then
$f$ is injective in $W$
How to prove this? If that inequality is right is it mean that the images are equal or not?
Assume $f(x) = f(y)$. What is $\|f(x)-f(y)\|$? What can you conclude about $\|x-y\|$?