This short guide is intended to help the reader to find his way about the book.

As we indicated in the Foreword, the book consists of a basic course on affine approximation, which we refer to colloquially as ‘the Dieudonné course’, linked at various stages with various geometrical examples whose construction is algebraic and to which the topological and differential theorems are applied.

Chapters 1 and 2, on sets, maps and the various number systems serve to fix basic concepts and notations. The Dieudonné course proper starts at Chapter 3. Since the intention is to apply linear algebra in analysis, one has to start by studying linear spaces and linear maps, and this is done here in Chapters 3 to 7 and in the first part of Chapter 8. Next one has to set up the theory of topological spaces and continuous maps. This is done in Chapter 16, this being prefaced, for motivational and for technical reasons, by a short account of normed linear spaces in Chapter 15. The main theorems of linear approximation are then stated and proved in Chapters 18 and 19, paralleling Chapters 8 and 10, respectively, of Prof. Dieudonné's book.

The remainder of the book is concerned with the geometry. We risk a brief consideration of the simplest geometrical examples here, leaving the reader to come back and fill in the details when he feels able to do so.