Using MathSciNet, I found the most cited book in each of 44 “pure” areas of Mathematical Subject Classification, from General (00) to Probability (60). The results are below. Bold font indicates I own the book. Italic: used to own, but not anymore.

Area | Title | Authors(s) |
---|---|---|

General | Real and complex analysis | Rudin |

History and biography | A course of modern analysis. | Whittaker, Watson |

Mathematical logic and foundations | Classical descriptive set theory | Kechris |

Combinatorics | Symmetric functions and Hall polynomials | Macdonald |

Order, lattices, ordered algebraic structures | Lattice theory | Birkhoff |

General algebraic systems | A course in universal algebra | Burris, Sankappanavar |

Number theory | The arithmetic of elliptic curves | Silverman |

Field theory and polynomials | Local fields | Serre |

Commutative algebra | Commutative algebra | Eisenbud |

Algebraic geometry | Algebraic geometry | Hartshorne |

Linear and multilinear algebra | Matrix analysis | Horn, Johnson |

Associative rings and algebras | Hopf algebras and their actions on rings | Montgomery |

Nonassociative rings and algebras | Infinite-dimensional Lie algebras | Kac |

Category theory; homological algebra | Categories for the working mathematician | MacLane |

K-theory | Cyclic homology | Loday |

Group theory and generalizations | Endliche Gruppen | Huppert |

Topological groups, Lie groups | Groupes et algèbres de Lie | Bourbaki |

Real functions | Convex analysis | Rockafellar |

Measure and integration | Geometric measure theory | Federer |

Functions of a complex variable | Bounded analytic functions | Garnett |

Potential theory | Nonlinear potential theory of degenerate elliptic equations | Heinonen, Kilpeläinen, Martio |

Several complex variables and analytic spaces | Function theory in the unit ball of C^{n} |
Rudin |

Special functions | Special functions | Andrews, Askey, Roy |

Ordinary differential equations | Introduction to functional-differential equations | Hale, Verduyn Lunel |

Partial differential equations | Elliptic partial differential equations of second order | Gilbarg, Trudinger |

Dynamical systems and ergodic theory | Random dynamical systems | Arnold |

Difference and functional equations | Stability of functional equations in several variables | Hyers, Isac, Rassias |

Sequences, series, summability | Divergent Series | Hardy |

Approximations and expansions | Constructive approximation | DeVore, Lorentz |

Harmonic analysis on Euclidean spaces | Harmonic analysis | Stein |

Abstract harmonic analysis | Abstract harmonic analysis | Hewitt, Ross |

Integral transforms, operational calculus | The classical moment problem and some related questions in analysis | Akhiezer |

Integral equations | Evolutionary integral equations and applications | Prüss |

Functional analysis | Sobolev spaces | Adams |

Operator theory | Semigroups of linear operators and applications to partial differential equations | Pazy |

Calculus of variations and optimal control | Optimization and nonsmooth analysis | Clarke |

Geometry | Projective geometries over finite fields | Hirschfeld |

Convex and discrete geometry | Convex bodies: the Brunn-Minkowski theory | Schneider |

Differential geometry | Differential geometry, Lie groups, and symmetric spaces | Helgason |

General topology | General topology | Engelking |

Algebraic topology | Algebraic topology | Spanier |

Manifolds and cell complexes | Introduction to compact transformation groups | Bredon |

Global analysis, analysis on manifolds | Introduction to the modern theory of dynamical systems | Katok, Hasselblatt |

Probability theory and stochastic processes | Convergence of probability measures | Billingsley |