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, £Γ/)-manifold pairs (M, N) (with M an iZ-manif old and N an ϋΓ/-manifold lying in M so that each coordinate chart f of M may be taken to be a homeomorphism of pairs (U, UnN) -^ (/(ED, ΛU) n Hf)).This implies that in the countably infinite Cartesian product of H with itself, the infinite (weak) direct sum of Hf with itself is homeomorphic to Hf (the two form such a pair), and that if K is a locally finite-dimensional simplicial complex equipped with the barycentric metric (inducing the Euclidean metric on each simplex) and if no vertex-star of K contains more than dim (H) vertices, then (K X H, K X Hf) is an (H 9 £Z/)-manifoId pair. U U(x, m)(zX) U(x, m) U V(x, n)m=0 m = 0 c U C7(x, m) U V(x, n -1) c c U(x, 0) U V(x, 1) C 0),

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF