 Smallest angle between two subspaces

For two finite-dimensional subspaces with projections $P,Q$, define $\cos \vartheta ={\sup }_{\Vert u\Vert =\Vert v\Vert =1}|⟨u,v⟩|=\Vert PQ\Vert$, with vectors drawn from the respective subspaces. The angle is zero when they intersect nontrivially. To measure separation after removing a common intersection, restrict both subspaces to its orthogonal complement. The resulting angle controls sums of positive operators in the Kitaev geometrical lemma.