Question on Rudin 10.5

I don't understand what does it mean that "A primitive mapping is thus one that changes at most once coordinate" ( what do we mean in darkened $x$ in the $10.5$ ? is it the set of all points $x$ $\in$ $E$ where $g(x)$ $\neq$ $0$ ? )

Hence, I don't understand from where does the $(10)$ inequality come. $G(x) = x + [g(x) - x_m]e_m$.

Any help would be appreciated.

Solution 1:

The notation $\mathbf x$ stands for a vector in $\Bbb R^n$. Suppose, say, that $n=4$ and that $m=3$. Then$$G(x_1,x_2,x_3,x_4)=\bigl(x_1,x_2,g(x_3),x_4\bigr).$$So, only the third coordinate of $(x_1,x_2,x_3,x_4)$ is changed. And then$$G(x_1,x_2,x_3,x_4)=(x_1,x_2,x_3,x_4)+\bigl(0,0,g(x_3)-x_3,0\bigr).$$

Solution 2:

You should understand that i$\neq m $ doesn't mean that i $\lt m $