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I am looking for reference for following lemma.

Consider a set of hyperplanes $H$ in a N-dimensional Euclidean space. Let $S$ be the set of intersections of all elements of $H$. Take a point $w$. Repeat as follows: project $w$ on a hyperplane in $H$. If you repeat this, you will get closer to $S$. If you use all hyperplanes infinitely often, you converge to $S$.

In what books I can find such sort of results?

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  • $\begingroup$ Sounds like the method of alternating projections. $\endgroup$
    – Chris Godsil
    Commented Jan 25, 2013 at 12:53

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Is this what you want?

Halperin, Israel The product of projection operators. Acta Sci. Math. (Szeged) 23 1962 96–99.

MathSciNet review: It is proved that if E1,E2,⋯,En are projections in a Hilbert space H and if T=E1E2⋯En, then Tm converges strongly as m→∞ to the projection E whose range is ⋂ni=1(EiH). For n=2 this was discovered by J. von Neumann [Ann. of Math. (2) 50 (1949), 401–485, p. 475; MR0029101 (10,548a)]. For general n, F. E. Browder [J. Math. Mech. 7 (1958), 69–80; MR0092070 (19,1057a)] says that weak convergence of Tm to E was previously proved by S. Kakutani; Browder modified Kakutani's result to allow an infinite sequence (En)n≥1 of projections.

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