Normal operator iff norm on v equivalent to that of adjoint I am reading a proof of the proposition that 
an operator $T \in \mathcal{L}$ is normal iff 
$\lVert Tv\rVert = \lVert T^{*}v\rVert$ 
for all $v \in V$, where $T^{*}$ is the adjoint of $T$. 
The proof is conducted as follows: 
$$\begin{align*}
T\text{ is normal}&\iff T^*T-TT^* = 0\\
&\iff \langle (T^*T-TT^*)v,v\rangle = 0\\
&\iff \langle T^*Tv,v\rangle = \langle TT^*v,v\rangle\\
&\iff \lVert Tv\rVert^2 = \lVert T^*v\rVert^2.
\end{align*}$$
I am having difficulty understanding why the second and fourth lines follow from their predecessors. It seems that the second would only follow if $T$ is self-adjoint.
 A: 
$T\in\mathcal{L}(V)$ is a normal operator iff $T^*T= TT^*$

$\begin{align*}T\text{ is normal}&\implies T^*T=TT^*\\ 
&\implies \langle (T^*T-TT^*)v,v\rangle = 0\\
&\implies \langle T^*Tv,v\rangle - \langle TT^*v,v\rangle=0\\
&\implies \langle Tv, Tv\rangle =\langle T^*v,T^*v\rangle\\&\implies\lVert Tv\rVert^2 = \lVert T^*v\rVert^2
\end{align*}$
$\begin{align*}
\lVert Tv\rVert^2 = \lVert T^*v\rVert^2&\implies\langle Tv, Tv\rangle =\langle T^*v,T^*v\rangle
\\&\implies \langle T^*Tv,v\rangle - \langle TT^*v,v\rangle=0\\&\implies \langle (T^*T-TT^*)v,v\rangle = 0 \tag 1\\&\implies T^*T=TT^*\\&\implies T \text{ is normal}
\end{align*}$
$(1)$ $T^*T-TT^* \in\mathcal{L}(V) $ is self-adjoint.Hence $\langle (T^*T-TT^*)v,v\rangle = 0 \quad\forall v\in V$ implies $T^*T-TT^*=0$ .
Note: Let $T\in\mathcal(V, W) $ then  $T^*\in L(W, V) $ defined by $\langle Tv, w\rangle =\langle v, T^*w\rangle$.
Now it's easy to see that for $T\in\mathcal{L}(V) $ , $\langle T^*Tv, v\rangle =\langle Tv, (T^*)^*v\rangle=\langle Tv, Tv\rangle$
Similarly $\langle TT^*v, v\rangle =\langle T^*v, T^*v\rangle$
