Maximal 𝑅-sets, Grassmann spaces, and Stiefel spaces of a Hilbert space

Abstract

R-set. The R-sets are subsets of the metric space of all closed linear subspaces of a Hilbert space, which satisfy a condition related to the metric. For the explicit definition we refer to Chapter 4. Each R-set is contained in a uniquely determined smallest maximal R-set. The algebraic objects attached to an R-set depend on this maximal R-set only, and its structure is of crucial importance. The path components of a maximal R-set are simply Grassmann manifolds for a finite-dimensional Hilbert space. In the following we present a general theory of Grassmann and Stiefel manifolds associated with a maximal R-set of a Hilbert space. This theory contains many important features of the ordinary theory of Grassmann and Stiefel manifolds of a finite-dimensional Hilbert space. Our constructions and arguments are independent of the dimension, and the ordinary theory for finite-dimensional Hilbert spaces and its direct generalization to Hilbert spaces of arbitrary dimension appear as special cases. We show that the path components of a maximal R-set are Banach manifolds and homogeneous spaces determined by the action of the group of continuous isomorphisms of the Hilbert space onto itself which leave the maximal R-set invariant. Furthermore, there is a natural fibre bundle structure. Several other Banach manifolds are associated in a canonical way with a path component of a maximal R-set. They are homogeneous spaces with a fibre bundle structure and direct analogues of the ordinary Stiefel manifolds. Finally, we investigate the maximal R-set of all closed linear subspaces of a