= Directed subspace angle
{title2=$\cos\theta_{V,W}=\inf_{v\in V,\ \|v\|=1}\|P_Wv\|$}
For a nonzero <closed subspace of a Hilbert space> $V$ and a <closed subspace of a Hilbert space> $W$, this angle measures the worst loss under projection from $V$ to $W$. Its positive cosine is a uniform lower bound for $P_W|_V$. The order of the subspaces matters in general. This is distinct from the <smallest angle between two subspaces>, whose cosine is a supremum. For equal finite dimensions the cosine is the smallest <singular value> of the cross <Gram matrix> of <orthonormal bases>. When $V$ is zero, the infimum is over an empty unit sphere and does not define an angle in $[0,\pi/2]$; use a lower-bound formulation instead.
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