This balanced introduction covers all fundamentals, from the real number system and point sets to set theory and metric spaces. Useful references to the literature conclude each chapter. 1956 edition.

This is an English translation of Bourbaki’s Fonctions d'une Variable Réelle. Coverage includes: functions allowed to take values in topological vector spaces, asymptotic expansions are treated on a filtered set equipped with a comparison scale, theorems on the dependence on parameters of differential equations are directly applicable to the study of flows of vector fields on differential manifolds, etc.

This text is for a beginning graduate course in real variables and functional analysis. It assumes that the student has seen the basics of real variable theory and point set topology. Contents: 1) The topology of metric spaces. 2) Hilbert Spaces and Compact operators. 3) The Fourier Transform. 4) Measure theory. 5) The Lebesgue integral. 6) The Daniell integral. 7) Wiener measure, Brownian motion and white noise. 8) Haar measure. 9) Banach algebras and the spectral theorem. 10) The spectral theorem. 11) Stone's theorem. 12) More about the spectral theorem. 13) Scattering theory.

This textbook leads the reader by easy stages through the essential parts of the theory of sets and theory of measure to the properties of the Lebesgue integral. The first part of the book gives a general introduction to functions of a real variable, measure, and integration, while the second part treats the problem of inverting the derivative of continuous functions, leading to the Denjoy integrals, and studies the derivates and approximate derivates of functions of a real variable on arbitrary linear sets. The author considers the presentation of this second part as the main purpose of his book.

This systematic exposition outlines the fundamentals of the theory of single sheeted domains of holomorphy. It further illustrates applications to quantum field theory, the theory of functions, and differential equations with constant coefficients. Students of quantum field theory will find this text of particular value. The text begins with an introduction that defines the basic concepts and elementary propositions, along with the more salient facts from the theory of functions of real variables and the theory of generalized functions. Subsequent chapters address the theory of plurisubharmonic functions and pseudoconvex domains, along with characteristics of domains of holomorphy. These explorations are further examined in terms of four types of domains: multiple-circular, tubular, semitubular, and Hartogs' domains. Surveys of integral representations focus on the Martinelli-Bochner, Bergman-Weil, and Bochner representations. The final chapter is devoted to applications, particularly those involved in field theory. It employs the theory of generalized functions, along with the theory of functions of several complex variables.