The ambient homeomorphy of an incomplete subspace of infinite-dimensional Hilbert spaces

Abstract

The pair (H, Hf) is studied from a topological point of view (where H is an infinite-dimensional Hilbert space and Hf is the linear span in H of an orthonormal basis), and a complete characterization is obtained of the images of Hf under homeomorphisms of H onto itself. As the characterization is topological and essentially local in nature, it is applicable in the context of Hilbert manifolds and provides a characterization of (if, /)-manifold pairs (M, N) (with M an iZ-manif old and N an /-manifold lying in M so that each coordinate chart f of M may be taken to be a homeomorphism of pairs