< Angle

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Here are some thoughts about angles in complex Hilbert spaces. I moved them from the main page because they don't qualify as encyclopedic knowledge. One could also use the absolute value of the dot product I suppose. --AxelBoldt

For complex Hilbert spaces, the formula (*) can be recycled to obtain a complex angle, but it is not entirely clear that this corresponds to a real-world notion of angle. An alternative is to use

(**) R(u·v)=cosθ ||u|| ||v||

where R denotes the real part. Definition (**) also special cases to (*) for real Hilbert spaces, so that may be a reasonable choice.

The definition of angle on the main page seems rather vague. Perhaps a better definition would be: the fraction of the arc of a circle with a center at the origin of the angle.

That way degrees can be clearly defined as:


and radians as:

(s/c)×2π = s/r

where s = arc length, c = circumphrence, and r = radius.