Projective Spaces

Abstract

The partially-ordered system of subspaces of a projective space form a matroid, thus endowing every projective space with a “projective dimension” (one less than the associated matroid rank). The goal of the chapter is a complete proof of the Veblen–Young theorem, which asserts that every projective space whose projective dimension exceeds two is the classical geometry of one- and two-dimensional subspaces of a (possibly infinite-dimensional) right vector space over a division ring. The so-called “Fundamental Theorem of Projective Geometry” asserts that any embedding of a classical projective space into another is induced by a semilinear transformation of the underlying vector spaces. It is proved here, allowing infinite dimensional spaces. The chapter concludes with three technical results that are vital to the classification of the polar spaces of the next chapter.