Q) Let $(e_n)$ be an orthonormal basis for a Hilbert space $H$. Given a sequence $(\alpha_n) \subset \mathbb{R}$, let
$$x_n = \cos\alpha_ne_{2n}-\sin\alpha_ne_{2n+1}, x_{2n+1} = \sin\alpha_ne_{2n}+\cos\alpha_ne_{2n+1}$$
Prove that $(x_n)$ is also an orthonormal basis of $H$.
I can see that $(x_n)$ is orthonormal but how can I show that it is a basis for $H$?
Best Answer
$cos(\alpha_n)x_n+sin(\alpha_n)x_{2n+1}=e_{2n}$, $-sin(\alpha_n)x_n+cos(\alpha_n)x_{2n+1}=e_{2n+1}$.