Preface

I. 
Motivation 

I.1 
The ThreeDimensional 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 Semilinear 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; Semibilinear Forms 

IV.2 
Null Systems 

IV.3 
Representation of Polarities 

IV.4 
Isotropic and Nonisotropic 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 Nonisotropic 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 Antiisomorphisms of the Endomorphism Ring 


Appendix I. 
The Twosided 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 Semilinear Transformations 

VI.7 
The Isomorphisms of the Group of Semilinear 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 