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, ${}\varphi$ 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 ${}E^{\perp }$ is also, due to fact, ${}\varphi$-invariant, and the restriction

$\varphi {|}_{E^{\perp }}\colon E^{\perp }\longrightarrow E^{\perp }$

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

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