The Method of Alternating Projections

TL;DR

This paper reviews the method of alternating projections in Hilbert spaces, providing proofs of convergence in norm and weak topology, including a new proof for the two-subspace case, and discusses a counterexample for norm convergence.

Contribution

It offers simplified proofs of known convergence results and introduces a novel proof for von Neumann's theorem on two subspaces, along with a counterexample for norm convergence.

Findings

Sequence converges in norm for periodic projections

Sequence converges in weak topology regardless of order

Counterexample shows lack of norm convergence in some cases

Abstract

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.