标题： 大多数投影迭代收敛 显示英文标题

标题： Most Iterations of Projections Converge

摘要： 考虑一个希尔伯特空间 $H$ 中的三个闭线性子空间 $C_1, C_2,$ 和 $C_3$ 以及它们的正交投影 $P_1, P_2$ 和 $P_3$。 Halperin 表明，可以通过周期性地将任意点 $x_0 \in H$ 投影到所有集合上来找到 $C_1\cap C_2 \cap C_3$ 中的一个点。 然后，极限点是$x_0$在$C_1\cap C_2 \cap C_3$上的投影。 然而，非周期的投影顺序可能导致投影级数不收敛，如 Kopecká、Müller 和 Paszkiewicz 所示。 这引发了这样一个问题：在$\{1,2,3\}^\mathbb{N}$中有多少个投影顺序是“行为良好”的，即它们导致收敛的投影级数。 Melo、da Cruz Neto 和 de Brito 提供了一个投影级数收敛的充要条件，并表明“行为良好”的投影顺序在具有全积测度的意义上形成一个大的子集。 我们证明从拓扑观点来看，“行为良好”的投影顺序的集合也是一个大的子集：在积拓扑下，它包含一个稠密的$G_\delta$子集。 此外，我们分析了为什么测度论情况下的证明不能直接适应到拓扑设置中。 显示英文摘要

摘要： 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.