Reflections - Kane's Book

first of all, realize that $\phi(H_\alpha) = H_{\phi(\alpha)}$ is an equality of two sets.

so suppose $y \in \phi(H_\alpha)$ . then $y = \phi(x)$ for some $x \in H_\alpha$ .

so $(x,\alpha) = 0$ and because $\phi$ is orthogonal, $(\phi(x),\phi(\alpha)) = 0$ , that is $(y,\phi(\alpha)) = 0$

so $y \in H_{\phi(\alpha)}$ , so $\phi(H_\alpha) \subseteq H_{\phi(\alpha)}$ .

now suppose we have $z \in H_{\phi(\alpha)}$ , so that $(z,\phi(\alpha)) = 0$ .

since $\phi$ is an orthogonal automorphism, it is bijective, and its inverse $\phi^{-1}$ is also orthogonal.

so $(\phi^{-1}(z),\phi^{-1}(\phi(\alpha)) = (z,\phi(\alpha)) = 0$ . but $(\phi^{-1}(z),\phi^{-1}(\phi(\alpha)) = (\phi^{-1}(z),\alpha)$ .

this means that $\phi^{-1}(z) \in H_\alpha$ so $z = \phi(\phi^{-1}(z)) \in \phi(H_\alpha)$ ,

which shows that $H_{\phi(\alpha)} \subseteq \phi(H_\alpha)$ , thus the two sets are equal.

the second equality, is an equality of functions, so we must show that the two functions have the same value at every point of $E$ .

so let $v \in E$ . since $\phi$ is surjective, we can write $v = \phi(w)$ for some $w = x + k\alpha$ ,

where $x \in H_\alpha ,k \in \mathbb{R}$ . so then we have:

$\phi s_\alpha \phi^{-1}(v) = \phi s_\alpha \phi^{-1}(\phi(w)) = \phi s_\alpha(w) = \phi s_\alpha(x + k\alpha) = \phi(s_\alpha(x) + ks_\alpha(\alpha))$

$= \phi(x - k\alpha) = \phi(x) - k\phi(\alpha) = s_{\phi(\alpha)}(\phi(x) + k\phi(\alpha)) = s_{\phi(\alpha)}(\phi(x+k\alpha))$

$= s_{\phi(\alpha)}(\phi(w)) = s_{\phi(\alpha)}(v)$ for all $v \in E$ .

this shows the two functions are indeed equal on all of $E$ .