Iterated Products of Projections in Hilbert Space.

The article discusses the theorem on convergence of the iterated product of orthogonal projections in Hilbert space. The theorem which resulted from I. Halperin's proof for an arbitrary finite number of projections is presented. Proofs for this theorem, including that of H. H. Bauschke, et. al., I. Amemiya and T. Ando, K. T. Smith, et. al., and N. Nakano, are mentioned. S. Kakutani's lemma and proof of the theorem are presented. Another theorem about nonexpansive, nonnegative operator on the Hilbert space is also proven.