Subspaces in Hermitian spaces of countable dimension. II
This is the second part of an investigation of subspaces in Hermitian spaces of countable dimension over \({\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{H}}\). In the first part [Math. Z. 184, 61-96 (1983; Zbl 0517.46016] we have proved a congruence theorem that classifies an extensive class of subspaces modulo the action of the orthogonal group of the whole space E. In the present paper this congruence theorem is supplemented by the results concerning restrictions on the invariants, i.e. concerning the existence of subspaces with prescribed invariants, and it is applied to subspaces of special interest: The discussion includes \(\perp\)-closed subspaces (\(\perp\)-dense subspaces are discussed in part I), hyperbolic subspaces, subspaces of finite codimension, and maximal positive definite subspaces. E.g., if E is of finite Witt index n then there are \((1/6)(n+1)(n+2)(n+3)\) orbits of maximal positive definite subspaces, one of them consisting of orthogonal summands, and \(n+1\) orbits consisting of \(\perp\)-dense subspaces. We also apply the methods of part I to the problem of extending given isometries between subspaces to isometries of E and, finally, we indicate the extent to which our results generalize to orderable base fields other than \({\mathbb{R}}\).
scientific article; zbMATH DE number 3847473