A gentle guide to the basics of two projections theory

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Isometries of Grassmann spaces (2015, background): “✷ The following lemma is known as the two projections theorem (see [3, 7]) in the special case when P,Q ∈ Pn(H) and ‖P − Q‖ < 1.”
Products of orthogonal projections and polar decompositions (2010, methodology): “As observed by the referees, the techniques of Dixmier, Afriat, Davis and Halmos, as recently surveyed by Galántai [17] and Böttcher and Spitkovsky [7], can be used to prove most of our results.”