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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.
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