A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines

Abstract

In this paper we give a complete and self-contained proof that real Hubert space, k, is homeomorphic to the countable infinite product of lines, s; symbolically h~s. We assume only that the reader understands the material of a first year course in topology: for example, elementary notions of complete metric spaces, product spaces, continuity, homeomorphisms, and the Tietze Extension Theorem. While the treatment is elementary, the arguments are not necessarily simple. The reader should be prepared to draw figures and verify continuity statements.