This classic and long out of print text by the famous French mathematician Henri Cartan, has finally been reissued as an unabridged reprint of the Kershaw Publishing Company 1971 edition at remarkably low price for a new generation of university students and teachers. It provides a concise and beautifully written course on rigorous analysis. Unlike most similar texts, which usually develop the theory in either metric or Euclidean spaces, Cartan's text is set entirely in normed vector spaces, particularly Banach spaces. This not only allows the author to develop carefully and geometrically the concepts of calculus in a setting of maximal generality, it allows him to unify the theory of both single and multivariable calculus over either the real or complex scalar fields by considering derivatives as linear transformations. More importantly, its republication in an inexpensive edition finally makes available again the English translations of both long-separated halves of Cartan’s famous 1965-6 calculus course at the University of Paris: The second half has been in print for over a decade as Differential Forms, published by Dover Books. Without the first half, it has been very difficult for readers of that second half text to be prepared with the proper prerequisites as Cartan originally intended. With both texts now available at very affordable prices, the entire course can now be easily obtained and studied as it was originally intended. The publisher, Karo Maestro, has written a lengthy preface and supplemental bibliography to provide historical and academic context for the book’s suggested use as a class text or self-study with its sequel. He also gives detailed recommendations of additional references for further study in analysis. The book is divided into two chapters. The first develops the abstract differential calculus. The introductory section provides an overview of the algebra and topology of Banach spaces, including norms, metrics, completeness, limits, convergence, isomorphisms and dual spaces along with important examples. An introduction to multilinear algebra is given via the exterior product and a brief digression into Banach algebras, thus setting the stage for the sequel. Then the Frechet derivative is defined and proofs are given of the two basic theorems of differential calculus: The mean value theorem and the inverse function theorem. The chapter proceeds with the algebra of polynomials in Banach spaces, the corresponding study of higher order derivatives and a proof of Taylor's formula. It closes with a study of local maxima and minima including both necessary and sufficient conditions for the existence of such minima. The second chapter is devoted to differential equations and really is the strength of the book. Indeed, it can easily be used for an honors course on differential equations that gives a completely rigorous treatment at the advanced undergraduate level. The general existence and uniqueness theorems for ordinary differential equations on Banach spaces are proved and applications of this material to linear equations and to obtaining various properties of solutions of differential equations are then given. Finally the relation between partial differential equations of the first order and ordinary differential equations is discussed and developed in detail. The prerequisites for this book are a rigorous first course in calculus using the ɛ-δ definitions of convergence and limits (or equivalently, a course in one variable advanced calculus or elementary analysis), a careful course in linear algebra on abstract vector spaces with norms and linear transformations as well as fluency with matrix computations and a basic course in differential equations. A knowledge of the computational aspects of multivariable calculus will also be needed for some parts of the book. The basic definitions of topology (metric and topological spaces, open and closed sets, etc.) will be needed as well.