We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself. Let $f \in O(E,q)$. We assume that for all $x \in E$ such that $q(x) \neq 0$, we have $f(x) - x \neq 0$ and $q(f(x)-x) = 0$. We propose to demonstrate that $f \in O^+(E,q)$ and that $E$ is an Artin space. a) Show that $\operatorname{dim}(E) \geq 3$. b) We denote by $V = \operatorname{Ker}(f - \operatorname{Id}_E)$. Show that $q_{/V} = 0$. c) Let $x \in E$ such that $q(x) = 0$. We denote $H = \{x\}^\perp$. Show that $q_{/H}$ is not identically zero. Deduce that there exists $y \in E$ such that $q(x+y) = q(x-y) = q(y) \neq 0$. d) We denote by $U = \operatorname{Im}(f - \operatorname{Id}_E)$. Show that $q_{/U} = 0$. e) Show that $U^\perp = V = U$. f) Deduce that $E$ is an Artin space and that $f \in O^+(E,q)$.
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $f \in O(E,q)$. We assume that for all $x \in E$ such that $q(x) \neq 0$, we have $f(x) - x \neq 0$ and $q(f(x)-x) = 0$.
We propose to demonstrate that $f \in O^+(E,q)$ and that $E$ is an Artin space.
a) Show that $\operatorname{dim}(E) \geq 3$.
b) We denote by $V = \operatorname{Ker}(f - \operatorname{Id}_E)$. Show that $q_{/V} = 0$.
c) Let $x \in E$ such that $q(x) = 0$. We denote $H = \{x\}^\perp$. Show that $q_{/H}$ is not identically zero. Deduce that there exists $y \in E$ such that $q(x+y) = q(x-y) = q(y) \neq 0$.
d) We denote by $U = \operatorname{Im}(f - \operatorname{Id}_E)$. Show that $q_{/U} = 0$.
e) Show that $U^\perp = V = U$.
f) Deduce that $E$ is an Artin space and that $f \in O^+(E,q)$.