Most Iterations of Projections Converge

Abstract

Consider three closed linear subspaces \(C_1, C_2,\) and \(C_3\) of a Hilbert space H and the orthogonal projections \(P_1, P_2\) and \(P_3\) onto them. Halperin showed that a point in \(C_1\cap C_2 \cap C_3\) can be found by iteratively projecting any point \(x_0 \in H\) onto all the sets in a periodic fashion. The limit point is then the projection of \(x_0\) onto \(C_1\cap C_2 \cap C_3\). 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 \(\{1,2,3\}^{\mathbb {N}}\) 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 \(G_\delta \) 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.