If T is a normal transformation, does that mean that $||v||^2=||Tv||^2$?

Question

If T is a normal transformation, does that mean that $||v||^2=||T||^2$ for all $v\in V$ ? Where $V$ is a vector space

And if yes, how to prove it?

EDIT: To be clear I mean that $||v||^2=(v,v)$

Answer 1

The usual meaning of "normal" is that you have an inner-product vector space where the norm is given by $\|v\|=\langle v,v\rangle^{1/2}$, and $T$ is normal if it commutes with its adjoint, i.e. $T^*T=TT^*$.

In the situation above, your equality fails often. For instance, take $Tv=2v$.