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Let $E$ be an Hilbert space and $F, G$ two subspaces such that $F \cap G =\{0\}$. Let $(x_n)$ be the sequence of iterated orthogonal projections: $x_0 \in F$, $x_1$ is the orthogonal projection of $x_0$ on $G$, $x_2$ the orthogonal projection of $x_1$ on $F$ and so on... We can suppose that $\|x_0\|=1$.

Does $(x_n)$ always have $0$ as limit? This is true if $E$ is of finite dimension. What about infinite dimension?

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  • $\begingroup$ This is a well studied topic. For the most recent results look at papers by Eva Kopecka. $\endgroup$
    – Bill Johnson
    Commented Nov 9, 2013 at 14:39
  • $\begingroup$ Also the case of subspaces replaced by convex sets (and also with more than two of them) is well studied. Buzzwords are "POCS" (projections onto convex sets), alternating projection method, von Neumann's alternating projection algorithm. In case of closed set you get at least weak convergence but probably that is not the case you are interested in? $\endgroup$
    – Dirk
    Commented Nov 9, 2013 at 16:19

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Thank you for your comments. The theorem 4.1 of article Alternating projection algorithm for two sets proves that the answer to the initial question is always positive. It also extends the question to convex sets.

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