= Real angle in a complex inner product space
{title2=$\cos\theta=\operatorname{Re}\langle u,v\rangle/(\|u\|\|v\|)$}
For nonzero <vectors> in a complex <inner product space>, the displayed quotient defines the ordinary real <angle> in the underlying real <inner product space>, whose <inner product> is $\operatorname{Re}\langle u,v\rangle$. The complex <Cauchy-Schwarz inequality> bounds it between $-1$ and $1$. This convention depends on the vectors' <phase factors>. The different phase-invariant convention $|\langle u,v\rangle|/(\|u\|\|v\|)$ measures an angle between complex one-dimensional subspaces; the two conventions should not be interchanged.
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