与非紧Stiefel流形相关的某些集合的道路连通性与拓扑闭包
Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold
AI总结：
本文研究非紧Stiefel流形相关集合的拓扑性质，证明了平移交集集合的折线道路连通性与交集集合拓扑闭包包含过流形内点的多项式路径，推动了非紧Stiefel流形拓扑研究及有限维连续希尔伯特空间框架理论发展。
摘要：
若$H$为希尔伯特空间，非紧Stiefel流形$St(n,H)$由$H$中线性无关的$n$元组构成。本文致力于非紧Stiefel流形的拓扑研究，主要证明了两个与非紧Stiefel流形相关集合的道路连通性及拓扑闭包的结果。第一部分中，在引入并证明一个关键引理后，我们证明：在平移$J$族各分量张成空间的余维数满足某一条件时，集合$\bigcap_{j \in J} \bigl( U(j) + St(n,H) \bigr)$可通过折线路径实现道路连通。第二部分中，我们证明$St(n,H) \cap S$的拓扑闭包包含所有含于$S$且经过$St(n,H)$中某一点的多项式路径。作为推论，我们证明$St(n,H)$在某一类子集中是相对稠密的，并通过框架理论中的诸多示例加以说明，这些示例源自对有限维连续框架对应的线性与二次方程解的研究。由于$St(n,L^2(X,μ;\fF))$与$\fF_{(X,Σ,μ),n}^\fF$等距同构，本文也对有限维连续希尔伯特空间框架理论作出了贡献。
英文摘要：
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 $\bigcap_{j \in J} \left( U(j) + St(n,H) \right)$ 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 $St(n,H) \cap S$ 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 $St(n,L^2(X,μ;\mathbb{F}))$ is isometric to $\mathcal{F}_{(X,Σ,μ),n}^\mathbb{F}$, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.
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