A series of textbooks for a reader working alone, from a first proof to the graduate qualifying exam in mathematics.
Each volume is a book of its own, with its own contents, search and exercises. A reference to another volume links straight to it, and hovering over the link shows the cited result without leaving the page.
Work in progress: these are drafts, rebuilt from the current branches, and chapters appear as they are written.
Where to start, and what each volume needs.
Start with F1 and read on to F2. Gate 0, the examination at the end of F2, Chapter 3, opens AN1. F3 builds
Proof, sets and numbers: what every later volume takes for granted, taught from the beginning.
F1
Logic as the grammar of proof, for a reader who has never been asked to write one: derivations, rule by rule, for the connectives and quantifiers; truth tables, soundness and completeness; and the way back from derivations to ordinary mathematical prose.
8 of 8 chapters written · 50 sections
F2
Sets and functions, relations and orders, induction and recursion, and the sizes of infinite sets, from diagonal arguments to the Schröder–Bernstein theorem. Chapter 3 ends with Gate 0, the examination to pass before analysis.
3 of 4 chapters written · 19 sections
F3
The number systems built out of sets:
7 of 7 chapters written · 41 sections
F4
The algebra the later volumes take for granted: the complex numbers built on
5 of 5 chapters written · 28 sections
Calculus made rigorous, then the analysis built on it: metric spaces, several variables, measure, Fourier and complex analysis.
AN1
Analysis on the real line, from the axioms of a complete ordered field: sequences and series, limits and continuity, the derivative, the Riemann–Darboux integral and power series, each proved from the least upper bound axiom.
1 of 11 chapters written · 7 sections
A companion series, written first and published on its own site; the volumes here cite it for facts about fields and matrices.
LA
Linear algebra for a reader working alone: vector spaces and linear maps, determinants and canonical forms, inner products and spectral theory, forms and tensors, and on to representations and numerical computation.
26 chapters · 237 sections
Mirrored here from its own site, linear-algebra.akizumifox.com, which holds the canonical copy.
The plan behind the series, kept as data.
Map
Every planned volume and chapter, and which chapters each one depends on, as an interactive graph.