In example 2.12 it is mentioned that the ellipsoid
$$\varepsilon = \{x \mid (x-x_c)^{T}P^{-1}(x-x_c)\leq 1\}$$
is the image of the unit euclidean ball $\{u \mid \|u\|_2\leq 1\}$ under the affine mapping $f(u) = P^{1/2} u + x_c$. What does this sentence mean? Does it mean that if we apply the affine mapping on the condition (that determines unit euclidean ball) we will get $\varepsilon$? But if it is right then after the application of affine mapping I get
$$\left\{ u \mid \sqrt{(P^{1/2}u+x_c)^{T}(P^{1/2}u+x_c)}\leq 1 \right\}$$
which does not look to be equal to the equation written above for $\varepsilon$. However, if I apply the affine mapping on the defining condition for $\varepsilon$ then I do get the unit euclidean ball. So based on this I conclude that the affine mapping should be applied to the defining condition of the set which is the image.
On the other hand when I read the solution for problem 2.17 of same book I see that the functional mapping (not necessarily affine) is applied on the set whose image is needed. So I am confused about what convention to follow here. I will be very thankful if somebody explains me where I am wrong. Thanks in advance.
If $u$ satisfies $\|u\|_2\leq 1$ then, as you can easily check, $x=f(u)=P^{1/2}u+x_c$ satisfies the condition mentioned in the definition of $\mathcal{E}$. That means that the image of the Euclidean ball under $f$ sits inside $\mathcal{E}$. It is not hard to find the inverse mapping.
I think the confusion arises because you went in the opposite direction with your substitutions.