Classical Geometries in Modern Contexts, 3rd Edition

The basic structure playing the key role in this book is a real inner product space (X, δ), i.e. a real vector space X together with a mapping δ : X × X → R, a so-called inner product, satisfying rules (i), (ii), (iii), (iv) of section 1 of chapter 1. In order to avoid uninteresting cases from the point of view of geometry, we will assume throughout the whole book that there exist elements a, b in X which are linearly independent.