= Solution
Over the <complex numbers>, the <fundamental theorem of algebra> makes every <irreducible polynomial> factor $f_j(x)$ equal to $x-\lambda_j$. Part (b)(iv) gives
$$
m_T(x)=\prod_{j=1}^r(x-\lambda_j)
$$
with distinct <eigenvalues>. The <primary decomposition theorem for linear maps> consequently expresses $V$ as the <direct sum> of the actual <eigenspaces> $E_{\lambda_j}=\ker(T-\lambda_jI)$, with no higher <generalized eigenspaces> needed. Part (ii) makes these summands mutually <orthogonal>.
Apply the <Gram-Schmidt process> to a <basis> within each <eigenspace>. Its <linear combinations> stay within that <eigenspace>, so all resulting <vectors> are still <eigenvectors>. Their union is <orthonormal> by part (ii) and spans $V$ by the <primary decomposition theorem for linear maps>. Thus
$$
\boxed{V=\bigoplus_{j=1}^{r} E_{\lambda_j},\qquad
[T]_{\mathcal B}=\operatorname{diag}(\lambda_1I_{\dim E_{\lambda_1}},\ldots,\lambda_rI_{\dim E_{\lambda_r}})}
$$
for an <orthonormal basis> $\mathcal B$ of <eigenvectors>. This proves the finite-dimensional complex <spectral theorem for normal operators> by the requested decomposition argument.
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