Vector difference norm bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$

Question

Given $\|x_1 - x_2 \| \leq C$ where C is a constant, could we derive a bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$?

Answer 1

There is a simple reason why you can't. Take two opposite vectors with arbitrary little norm. The difference between the two can be as small as you wish but the difference between their normalized counterparts will always be of norm 2.

For instance in $\mathbb R^2$ : $x1=(\frac 1 n,0),x_2=(-\frac 1 n,0),n\in \mathbb N$, the difference has norm $\vert\vert x_1-x_2\vert\vert=\frac 2 n$ but the difference between the normalized vector is 2.