2021/01/01 by Idrissi, Nizar El, Kabbaj, Samir, Moalige, Brahim
#42C15 #54D05 #54D99 #57N20 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)
paper · doi:10.48550/arxiv.2101.00322
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 \bigcapj ∈ J ( 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;\mathbbF)) is isometric to F(X,Σ,μ),n^\mathbbF, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.