First some notation/terminology. Let $T$ be some bounded linear operator acting on a Hilbert space $\mathcal{H}$. The null space is $\{x \in \mathcal{H} \mid Tx=0\}$ and the range space is the (norm?) closure of the set $\{Tx \mid x \in \mathcal{H}\}$. The corresponding projections are called the null projection, denoted by $N(T)$, and the range projection, $R(T)$.
Here is the theorem whose proof I'm having trouble following:
I'm just confused by this proof in general. Why does $\ker T = [T^*(\mathcal{H}]^{\perp}$ imply $N(T) = I - R(T^*)$? I'm pretty certain the square brackets [] mean norm closure. And why is $N(T) = N(T^*T)$? Does $\ker T = \ker T^*T$ imply this? Why?