= Solution
Let $u\in\ker(T-\lambda I)$ and $w\in\ker(T-\mu I)$, with $\lambda\ne\mu$. By part (i) and the <adjoint operator> identity,
$$
\lambda\langle u,w\rangle=\langle Tu,w\rangle
=\langle u,T^*w\rangle
=\langle u,\bar\mu w\rangle
=\mu\langle u,w\rangle.
$$
The last step uses conjugate-linearity in the second argument of the <inner product>. Therefore $(\lambda-\mu)\langle u,w\rangle=0$, proving the <orthogonality> of the two <eigenspaces>:
$$
\boxed{\ker(T-\lambda I)\perp\ker(T-\mu I)\quad(\lambda\ne\mu)}.
$$
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