Infinite-Dimensional Generalizations of Orthogonal Groups over Hilbert Spaces : Constructions and Properties

Abstract

In real Hilbert spaces, this paper generalizes the orthogonal groups $\mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $Θ(κ)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(κ)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$\mathrm{O}=\varinjlim\mathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $Θ(κ)$ is a topological and normal subgroup of $O(κ)$; (b) $O^{(n)}(κ) \to O^{(n+1)}(κ) \stackrelπ{\to} S^κ$ is a fibre bundle where $O^{(n)}(κ)$ is a subgroup of $O(κ)$ and $S^κ$ is a generalized sphere.