Euclidean space/Isometry/Structure/Fact/Proof

Proof

We do induction over the dimension ${}n$ of ${}V$. The one-dimensional case is clear, due to fact. Let ${}n=2$. The determinant has, because of fact either the value ${}1$ or ${}-1$. In case it is ${}-1$, the characteristic polynomial has two distinct zeroes, and these zeroes are, due to fact, ${}1$ and ${}-1$. Then we have a reflection at an axis, and

${}V=G\oplus H\,.$

If the determinant is ${}1$, then we are in the situation of fact, and we have a rotation. If the angle of rotation is ${}0$, then we have the identity, and we can decompose ${}V=G_{1}\oplus G_{2}$. If the angle of rotation is ${}\pi$, then we have a point reflection ${}-\operatorname {Id}$; therefore, we have a decomposition ${}V=H_{1}\oplus H_{2}$. For all other angles, there is no eigenvector.

Let now ${}n\geq 1$ be arbitrary, and suppose that the statement is proven for smaller dimensions. Because of fact, there exists a ${}\varphi$-invariant linear subspace ${}U$ of dimension ${}1$ or ${}2$, and, because of fact, there exists an invariant orthogonal complement, that is,

${}V=U\oplus W\,.$

The induction hypothesis, applied to ${}W$, yields the result.

To fact