On Polar Spaces of Infinite Rank

Abstract

Many properties of polar spaces of finite rank fail to hold in polar spaces of infinite rank. For instance, in a polar space of infinite rank it can happen that maximal singular subspaces have different dimensions; every polar space of infinite rank contains singular subspaces that cannot be obtained as intersections of any family of maximal singular subspaces, whereas in a polar space of finite rank every singular subspace is the intersection of a finite number of maximal singular subspaces. In this paper we shall examine peculiar properties of polar spaces of infinite rank, to test how far them can drug us from the familiar world of polar spaces of finite rank.