Text on Analysis on Banach Spaces: Differential Calculus On Normed Spaces by Henri Cartan!





      NEXT,Some Reviews: 

 “Written by the hand of a master in a nice and pleasant style, it is highly recommended to all those interested in the study of differential calculus……In conclusion, a wonderful geometrical (coordinate-free) introduction to the subject.”-M.Puta (Timişoara), Zentralblatt MATH 


……the book consists of two parts….. the first exposes the differential calculus in Banach spaces, and the ordinary differential equations. The second part develops, again on open sets in Banach spaces, the calculus of exterior forms, their integration in the finite dimensional case, and applications to variational calculus and to the differential geometry a la Eli Cartan of curves and surfaces in Euclidean space. In short, it is an excellent modern textbook on advanced analysis put in a non-standard form.“- I. Weisman,  Zentralblatt MATH 

       Further Reviews: 

Cartan’s work provides an excellent text for an undergraduate course in advanced calculus, but at the same time, it furnishes the reader with an excellent foundation for global and nonlinear algebra.” – Mathematical Review 


Brilliantly successful.” – Bulletin de l’Association des Professeurs de Mathematiques   


“The presentation is precise and detailed, the style lucid and almost conversational……clearly, an outstanding text and work of reference.” –Annales  

About Cartan’s Cours de calcul différentiel

    Cartan’s unique text has finally been reissued after nearly a half a century. It’s the 1st half of the English translation of Cartan’s famous Cours de calcul différentiel, given by the author at the University of Paris at the height of the Bourbaki movement in mathematics education. Bourbaki-the remarkable school of abstraction which the author helped co-found and popularize in Europe-probably had a deeper impact on the practice of both research and teaching mathematics in the 20th century then any other influence!  As a result, this course is a historical treasure as well as an academic one. 

          A Lost Classic Returns        

    The 1st text’s republication in a cheap edition finally makes readily available again the English translation of both long separated halves of Cartan’s famous 1965-6 analysis course at The University of Paris. The 2nd half has been in print for over a decade as Differential Forms, published by Dover Books.

           Without the 1st half, it has been very difficult for readers of that 2nd half to have the proper prerequisites. Consequently,this publication finally makes the entire course available again as Cartan originally intended!  


     Calculus Bourbaki Style 

    At the time of its publication, this presentation represented the height of rigor and abstract in a calculus course for talented students. This is due to the fact no undergraduate course had done calculus on abstract normed spaces. As a result, it’s like no other text on calculus/analysis at this level you’ll find anywhere.               

   What distinguishes Cartan’s course presented in this text & its sequel that they present careful calculus on Banach spaces.

    Cartan’s approach over normed instead of metric spaces has the main advantage of a unified theory of functions of one and several variables. For example, there is a single definition of a derivative as a linear transformation between subspaces of a Banach space. As a result, the needed vector space structure is already present.

  Therefore in the case of abstract metric spaces, one constructs this structure ad hoc.  As a result, Cartan derives special cases, such as partial derivatives and Taylor’s theorem, from the general case as needed.   

  As the title indicates, this text focuses solely on the foundations of differential calculus on Banach spaces. Consequently, the sequel presents in depth the integral calculus .

    Some Topics Covered  In The Text: 

  • ·      Linear algebra and topology in abstract normed spaces

  • ·       Multilinear continuous functions via the exterior product  (Therefore, a unified modern approach to multilinear algebra is presented.) 
  • ·      The Frechet derivative as a linear transformation and therefore in a coordinate free definition 
  • ·      The General Mean Value And Inverse Function Theorems 
  • ·      Taylor’s formula and higher order derivatives
  • ·      General existence and uniqueness theorems for linear homogeneous and nonhomogeneous ordinary differential equations
  • ·      Connection between solution spaces for partial differential equations and systems of ordinary differential equations
  • ……and much more!

    Prerequisites For The Course 

       The prerequisites for this text are:

   1)A rigorous first course in calculus/ elementary analysis using the ɛ-δ definitions of convergence and limits on the real line,


  2)A careful course in linear algebra on abstract vector spaces with norms and linear transformations as well as fluency with matrix computations and

  3) A basic computational course in differential equations. Also:    

  4) A knowledge of the computational aspects of multivariable calculus will also be needed for some parts of the book.

5) Finally, some familiarity with the basic definitions of topology (metric and topological spaces, open and closed sets, etc.) will be needed as well. 

               New Features Of The 2017 Edition

      Students studying this book will find a detailed new preface providing deep historical context and educational perspective on the author’s intentions with the book.     

       Finally, a detailed new bibliography by the publisher compares Cartan’s book(s) to the current standard analysis texts at this level. In addition, he offers suggestions for using Cartan’s book(s) in coursework, either as main text(s) or as supplements alongside a standard texts.   

    After the publication of this volume, you can buy the entire course very cheaply! Therefore, Cartan’s complete course can  become a standard analysis course for university students and teachers at this level!




Table of Contents

Preview page 30-41


              And Don’t Forget….

    The main purpose of printing this new edition of the first volume is so that both volumes of Cartan’s famous course can now be available readily and cheaply to English-speaking students and teachers of mathematics!

     As I said, the 2nd half been available from Dover Books as Differential Forms. The 2nd half is just as clear, beautifully written and informative as the first.Hence it completes the first edition. 


                 The General Differential Calculus       

    The text’s focus is on differential forms as tools in calculus on Banach spaces. The main use of forms is in path integrals in finite dimensional Banach spaces. In the first volume, the Frechet derivative generalizes and unifies all the special cases of differentiation in Euclidean spaces. Similarly, abstract path integrals over differential forms unify all the integrals of calculus as special cases of Stokes’ Theorem.Therefore, the 2 books together form a complete course in mathematical analysis on normed spaces!

  In addition,these methods are then applied to problems in classical differential geometry and the calculus of variations,making the book even more valuable.


In conclusion, since both books are available so cheaply now, there’s no reason not to have both.


(Ok, enough advertising for someone else.The fact is, God knows Dover can afford their own………………) 

        How Do I Get It? 

Click here to buy the Kindle E-Book.

 Here to buy either at Amazon.