This classic and long out of print text by the famous French mathematician Henri Cartan, has finally been retitled and 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 the concepts of calculus in a setting of maximal generality, it allows him to unify both single and multivariable calculus over either the real or complex scalar fields by considering derivatives of nth orders as linear transformations. This prepares the student for the subsequent study of differentiable manifolds modeled on Banach spaces as well as graduate analysis courses, where normed spaces and their isomorphisms play a central role. More importantly, it's republication in an inexpensive edition finally makes available again the English translations of both long separated halves of Cartan's famous 1965-6 analysis 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 book is divided into two chapters. The first develops the abstract differential calculus. After an introductory section providing the necessary background on the elements of Banach spaces, 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 introduction and 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. Then the general existence and uniqueness theorems for ordinary differential equations on Banach spaces are proved. 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. The prerequisites are rigorous first courses in calculus on the real line (elementary analysis), linear algebra on abstract vectors spaces with linear transformations and the basic definitions of topology (metric spaces, topology, etc.) A basic course in differential equations is advised as well. Together with its' sequel, Differential Calculus On Normed Spaces forms the basis for an outstanding advanced undergraduate/first year graduate analysis course in the Bourbakian French tradition of Jean Dieudonné's Foundations of Modern Analysis, but a more accessible level and much more affordable then that classic.
About the Author: About the Author: Henri Cartan ( 1904-2008) was a prominent French-born mathematician. He was the second generation to do so in his family-his father Eli was one of the founding fathers of modern differential geometry. Cartan was known as both a first rate researcher and an excellent teacher. Through his efforts in both, he helped revive research in the mathematical sciences in French academia in the generation between the World Wars. Cartan was a founding member of "The Bourbaki", a famous group of young French mathematicians whose tenant of completely general axiomatic foundations for mathematics in both research and teaching was highly influential in post-WW II Europe. Cartan taught at several prominent French universities, primarily at the University of Paris at Sorbonne. During his long career, he published significant papers on analytic functions, the theory of sheaves, homological theory, algebraic topology and potential theory. He also published several texts, including the seminal first book on the subject, Homological Algebra, with fellow Bourbakian Samuel Eilenberg. About the Publisher/ Editor: Karo Maestro aka The Mathemagician is the nome-de-plume of a former graduate student in mathematics. His true identity will remain a secret for now, but one day will be revealed to all. Some things this online enigma can reveal: He was a distinguished undergraduate student as a double major in mathematics and biochemistry whose poor health prevented completing graduate studies. Unbowed and undaunted, he plans to return to ultimately obtain a PhD before dying. Partially to that end, he is building a publishing company committed to making high quality sources of mathematics-both original works and reprints-available widely and inexpensively to students of all backgrounds. The volume you hold is the very first published. He has 3 bachelors' degrees, in philosophy, physical chemistry and mathematics as well as minors in biochemistry and psychology. He is also a past reviewer of textbooks for the Mathematical Association Of America. He's also the author of Tables, Chairs And Beermugs( https: //tableschairsandbeermugsmathemagician.blogspot.com/), the associated blog for the website where he reviews textbooks and vents on matters mathematical and academic. He's painfully blunt, opinionated and has a comment on just about everything in creation.