Published June 1, 2023 | Version v1
Journal article Open

Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold

Description

If H is a Hilbert space, the non-compact Stiefel manifold St(n,H) consists of independent n-tuples in H. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that jJ(U(j)+St(n,H)) is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating J-family. Then, in the second part, we show that the topological closure of St(n,H)S contains all polynomial paths contained in S and passing through a point in St(n,H). As a consequence, we prove that St(n,H) is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since St(n,L2(X,μ;F)) is isometric to F(X,Σ,μ),nF, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.

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path-connectedness-and-topological-closure-of-some-sets-related-to-the-non-compact-stiefel-manifold.pdf