Most Iterations of Projections Converge

Abstract

AbstractConsider three closed linear subspaces $$C_1, C_2,$$ C 1 , C 2 , and $$C_3$$ C 3 of a Hilbert space H and the orthogonal projections $$P_1, P_2$$ P 1 , P 2 and $$P_3$$ P 3 onto them. Halperin showed that a point in $$C_1\cap C_2 \cap C_3$$ C 1 ∩ C 2 ∩ C 3 can be found by iteratively projecting any point $$x_0 \in H$$ x 0 ∈ H onto all the sets in a periodic fashion. The limit point is then the projection of $$x_0$$ x 0 onto $$C_1\cap C_2 \cap C_3$$ C 1 ∩ C 2 ∩ C 3 . Nevertheless, a non-periodic projection order may lead to a non-convergent projection series, as shown by Kopecká, Müller, and Paszkiewicz. This raises the question how many projection orders in $$\{1,2,3\}^{\mathbb {N}}$$ { 1 , 2 , 3 } N are “well behaved” in the sense that they lead to a convergent projection series. Melo, da Cruz Neto, and de Brito provided a necessary and sufficient condition under which the projection series converges and showed that the “well behaved” projection orders form a large subset in the sense of having full product measure. We show that also from a topological viewpoint the set of “well behaved” projection orders is a large subset: it contains a dense $$G_\delta $$ G δ subset with respect to the product topology. Furthermore, we analyze why the proof of the measure theoretic case cannot be directly adapted to the topological setting.