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 $\Theta(\kappa)$, 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~$\mathrm{O}=\varinjlim\mathrm{O}(n)$ 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.