The Method of Alternating Projections

The method of alternating projections involves orthogonally projecting an element of a Hilbert space onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm if the projections are taken periodically, or even quasiperiodically. We present proofs of such well known results, and offer an original proof for the case of two closed subspaces, known as von Neumann's theorem. Additionally, it is known that this sequence always converges with respect to the weak topology, regardless of the order projections are taken in. By focusing on projections directly, rather than the more general case of contractions considered previously in the literature, we are able to give a simpler proof of this result. We end by presenting a technical construction taken from a recent paper, of a sequence for which we do not have convergence in norm.