Isometry/C/Diagonalizable/Fact/Proof

Proof

We do induction on the dimension of $V$ . The statement is clear in the one-dimensional case. Due to the Fundamental theorem of algebra and fact, $φ$ has an eigenvalue and an eigenvector, which we can normalize. Let $E \subseteq V$ be the corresponding eigenline. Since we have an isometry, the orthogonal complement $E^{⊥}$ is also, due to fact, $φ$ -invariant, and the restriction

φ |_{E^{⊥}} : E^{⊥} ⟶ E^{⊥}

is again an isometry. By the induction hypothesis, there exists on $E^{⊥}$ an orthonormal basis consisting of eigenvectors. Together with the first eigenvector, these form an orthonormal basis of eigenvectors of $V$ .

To fact