Exercise 7.B.4

Proof. The forward direction is basically the same as 7.22 and 7.24.

Suppose all pairs of eigenvectors corresponding to distinct eigenvalues of $T$ are orthogonal and

where ${\lambda }_1,\dots ,{\lambda }_m$ are distinct eigenvalues of $T$. Consider the constructed by concatenating orthonormal bases of $E({\lambda }_1,T),\dots ,E({\lambda }_m,T)$. By 5.41 (e) this list has length $\dim <!--\; FUNCTION\; APPLICATION\; -->V$. Moreover, because all pairs of eigenvectors corresponding to distinct eigenvalues of $T$ are orthogonal, by 6.26 this is list is also linearly indepedent. Therefore $V$ has an orthonormal basis consisting of eigenvectors of $T$ and is normal by 7.24. □