Vol. 34, No. 1, 1970

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

James Edward West

Vol. 34 (1970), No. 1, 257–267

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 (H,Hf)-manifold pairs (M,N) (with M an H-manifold and N an Hf-manifold lying in M so that each coordinate chart f of M may be taken to be a homeomorphism of pairs (U,U N) f(f(U),f(U) 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 ×H,K ×Hf) is an (H,Hf)-manifold pair.

Mathematical Subject Classification
Primary: 57.55
Received: 10 December 1969
Published: 1 July 1970
James Edward West