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

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 $Theta(kappa)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(kappa)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$mathrm{O}=varinjlimmathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $Theta(kappa)$ is a topological and normal subgroup of $O(kappa)$; (b) $O^{(n)}(kappa) to O^{(n+1)}(kappa) stackrel{pi}{to} S^{kappa}$ is a fibre bundle where $O^{(n)}(kappa)$ is a subgroup of $O(kappa)$ and $S^{kappa}$ is a generalized sphere.