Table of Contents

Preface I. Motivation I.1 The Three-Dimensional Affine Space as Prototype of Linear Manifolds I.2 The Real Projective Plane as Prototype of the Lattice of Subspaces of a Linear Manifold II. The Basic Properties of a Linear Manifold II.1 Dedekind's Law and the Principle of Complementation II.2 Linear Dependence and Independence; Rank II.3 The Adjoint Space Appendix I. Application to Systems of Linear Homogeneous Equations Appendix II. Paired Spaces II.4 The Adjunct Space Appendix III. Fano's Postulate III. Projectivities III.1 Representation of Projectivities by Semi-linear Transformations Appendix I. Projective Construction of the Homothetic Group III.2 The Group of Collineations III.3 The Second Fundamental Theorem of Projective Geometry Appendix II. The Theorem of Pappus III.4 The Projective Geometry of a Line in Space; Cross Ratios Appendix III. Projective Ordering of a Space IV. Dualities IV.1 Existence of Dualities; Semi-bilinear Forms IV.2 Null Systems IV.3 Representation of Polarities IV.4 Isotropic and Non-isotropic Subspaces of a Polarity; Index and Nullity Appendix I. Sylvester's Theorem of Inertia Appendix II. Projective Relations between Lines Induced by Polarities Appendix III. The Theorem of Pascal IV.5 The Group of a Polarity Appendix IV. The Polarities with Transitive Group IV.6 The Non-isotropic Subspaces of a Polarity V. The Ring of a Linear Manifold V.1 Definition of the Endomorphism Ring V.2 The Three Cornered Galois Theory V.3 The Finitely Generated Ideals V.4 The Isomorphisms of the Endomorphism Ring V.5 The Anti-isomorphisms of the Endomorphism Ring Appendix I. The Two-sided Ideals of the Endomorphism Ring VI. The Groups of a Linear Manifold VI.1 The Center of the Full Linear Group VI.2 First and Second Centralizer of an Involution VI.3 Transformations of Class 2 VI.4 Cosets of Involutions VI.5 The Isomorphisms of the Full Linear Group Appendix I. Groups of Involutions VI.6 Characterization of the Full Linear Group within the Group of Semi-linear Transformations VI.7 The Isomorphisms of the Group of Semi-linear Transformations VII. Internal Characterization of the System of Subspaces A Short Bibliography of the Principles of Geometry VII.1 Basic Concepts, Postulates and Elementary Properties VII.2 Dependent and Independent Points VII.3 The Theorem of Desargues VII.4 The Imbedding Theorem VII.5 The Group of a Hyperplane VII.6 The Representation Theorem VII.7 The Principles of Affine Geometry Appendix S. A Survey of the Basic Concepts and Principles of the Theory of Sets A Selection of Suitable Introductions into the Theory of Sets Sets and Subsets Mappings Partially Ordered Sets Well Ordering Ordinal Numbers Cardinal Numbers Bibliography Index