Subspaces in Hermitian spaces of countable dimension. II
type: Item
instance of: scholarly article
title: Subspaces in Hermitian spaces of countable dimension. II
zbMATH Open document ID: 0534.10013
DOI: 10.1007/BF01159177
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173540
review text: 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 \({\mathbb{R}}\).
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 3847473
zbMATH Keywords: subspaces
zbMATH Keywords: Hermitian spaces of countable dimension
zbMATH Keywords: congruence theorem
zbMATH Keywords: invariants
zbMATH Keywords: hyperbolic subspaces
zbMATH Keywords: finite codimension
zbMATH Keywords: maximal positive definite
zbMATH Keywords: isometries
author: Werner Baeni
MaRDI profile type: Publication
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OpenAlex ID: W2047446597
description: scientific article; zbMATH DE number 3847473