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In the context of the development of the spectral theory, J. v. Neumann showed that for every pair of closed linear subspaces $M,N$ of a Hilbert space the iterations $(P_M P_N)^nx$, where $P_M$ and $P_N$ is the orthogonal projection onto $M$ and $N$, respectively, converge to the projection onto the intersection $M\cap N$ of $M$ and $N$.

In their paper On the convergence of von Neumann's alternating projection algorithm for two sets H. H. Bauschke and J. M. Borwein intruduced the concept of regularities of tuples of closed and convex sets. The pair $(M,N)$ is called boundedly regular if for every bounded set $S$ and for every $\varepsilon>0$ there is a $\delta > 0$ such that $\max\{d(x,M), d(x,N)\} < \delta$ implies $d(x, M\cap N)<\varepsilon$ for $x\in S$.

This condition is shown to imply convergence of von Neumann's alternating projection method. In this paper they mention a pair of closed linear subspaces (with non-closed sum) for which they do not know whether it is boundely regular.

Is the answer to the question of whether every pair of closed linear subspaces is boundedly regular known?

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  • $\begingroup$ Take unit vectors $x_n,y_n$ in $H_n$ which are $1/n$-close but linearly independent and consider the $\ell_2$-direct sums $M$ of ${\bf C}x_n$ and $N$ of ${\bf C}y_n$. Then, $(M,N)$ in the $\ell_2$-direct sum $\bigoplus H_n$ is a counterexample. $\endgroup$ – Narutaka OZAWA Mar 14 at 4:38

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