= Close orthogonal projections identify their image bundles
{title2=$\|P-Q\|<1$}
Let P,Q be continuous orthogonal projection matrices of equal rank on a common trivial <complex vector bundle>, with the displayed bound at every base point. Restricting Q to the image of P is injective: if Qv=0, then $v=(P-Q)v$ contradicts the norm bound unless v=0. Equal dimensions give a continuously invertible <vector bundle isomorphism>. Compactness applies this observation uniformly to a homotopy of image planes.
Back to article page
