[Math] Proof of normal operator and self-adjoint operator

Question

1) Let $T∈L(V,V)$ be a normal operator. Prove that $||T(v)||=||T^*(v)||$ for every $v∈V$.
($T^*$ is the adjoint of $T$)

2) Let $T$ be an operator on the finite dimensional inner product space $(V,<,>)$ and assume that $TT^*=T^2$. Prove that T is self-adjoint. (Can I simple get $T=T^*$ from $TT^*=T^2$? So there is nothing to prove)

Thank you for this two questions.

Answer 1

1>

$||T(v)||^2 = <Tv,Tv> = <v,T^*Tv>=<v,TT^*v>=<T^*v,T^*v> =<T^*v,T^*v>=||T^*v||^2$