Terminology for vectors in ''positive angle'' position

Question

I would like to know whether there is a standard terminology for the following situation:

Let $H$ be a complex Hilbert space and $\xi, \eta \in H$ are two vectors such that $(\xi, \eta)_H \ge 0$. Do we say that $\xi$ and $\eta$ are in positive angle position ?

The same question for the following condition $(\xi, \eta)_H\in \mathbb{R}$, what is the terminology for it?

score 0 · Answer 1 · answered Jan 13 '15 at 22:18

0

Since for ${\Bbb{R}}^2$ and ${\Bbb{R}}^3$ (which are basis Hilbert spaces) we have $$(\xi,\eta)=||\xi||||\eta||\cos(\theta)$$ this gives

$(\xi,\eta)>0$, meanwhile $-\pi/2<\theta<\pi/2$,
$(\xi,\eta)<0$, meanwhile $\pi/2<\theta<3\pi/2$,
$(\xi,\eta)=0$, meanwhile $\theta=\pi/2,3\pi/2$.

does this help you to generalize?

answered Jan 13 '15 at 22:18

janmarqz

10,538

I forgot to say that $\theta$ is the angle between $\xi$ and $\eta$. – janmarqz Jan 13 '15 at 22:21
Thank you. But the main point here is that the Hilbert space is over the complex field $\mathbb{C}$. – Yanqi QIU Jan 14 '15 at 08:00
i know, but $\Bbb R^2\cong\Bbb C$ – janmarqz Jan 14 '15 at 14:58

Terminology for vectors in ''positive angle'' position

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