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 Kopeck\'{a}, M\"{u}ller, 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 from the measure theoretic case cannot be directly adapted to the topological setting.