T. M. Flett was a Professor of Pure Mathematics at the University of Sheffield from 1967 until his death in 1976. This book, which he had almost finished, has been edited for publication by Professor J. S. Pym. This text is a treatise on the differential calculus of functions taking values in normed spaces. The exposition is essentially elementary, though on are occasions appeal is made to deeper results. The theory of vector-valued functions of one real variable is particularly straightforward, and this forms the substance of the initial chapter. A large part of the book is devoted to applications. An extensive study is made of ordinary differential equations. Extremum problems for functions of a vector variable lead to the calculus of variations and general optimisation problems. Other applications include the geometry of tangents and the Newton-Kantorovich method in normed spaces. The three historical notes show how the masters of the past (Cauchy, Peano...) created the subject by examining in depth the evolution of certain theories and proofs.
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Book Description
T. M. Flett was a Professor of Pure Mathematics at the University of Sheffield from 1967 until his death in 1976. This book, which he had almost finished, has been edited for publication by Professor J. S. Pym. This text is a treatise on the differential calculus of functions taking values in normed spaces. The exposition is essentially elementary, though on are occasions appeal is made to deeper results.
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1 of 1 people found the following review helpful:
5.0 out of 5 stars
Every aspiring analyst should read it,
By JL "humblelearner" (Spain) - See all my reviews
This review is from: Differential Analysis: Differentiation, Differential Equations and Differential Inequalities (Paperback)
This book deals with some foundational properties of real differential calculus (one or several variables). It is not comprehensive but it is a model to follow and admire (since T. M. Flett died well before his work was finished and edited by J. M. Pym). It reminds me of the more elaborate Burckel's treatise on complex analysis An Introduction to Classical Complex Analysis: Volume 1 (Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften / Mathematische Reihe) (v. 1). Its main purpose is not empty erudition, but rather a quest for unveiling what the main concepts and theorems mean. This work digs out the truth from the methods that inspired some of the creators of differential calculus: Lagrange, Cauchy, Peano, Lipschitz, Weierstrass, Dini, Arzela, Fréchet, Picard, Perron, Osgood... Technical discussions of old proofs or attempts of proofs, are amazing and brings mathematics back to life. Unfortunately, the book is too short and somewhat irregular, since, it covers (in five chapters) only functions of one real variable, (real) ordinary differential equations (ODE), and Fréchet, Gâteux and Hadamard basic differential calculus on a topological vector spaces. I wasn't even aware of the existence of Hadamard differential, and my knowledge of Gâteaux differential is quite shallow, so I cannot express but a naive appreciation of both. But, Fréchet calculus on Banach spaces, as well as ODE and one variable calculus is so universally widespread now, that even I know something about that, and I have really enjoyed the first three chapters. Great mathematics, deep insight, a book that opens our mind and improve our previous background on the subject. I'm not a pro, but I'm sure learning (and teaching) mathematical analysis could be much easier if teachers had a serious historical background and a more critical view about the meaning and the technicalities of most basic results. One example? Read Flett's proof of Rolle's theorem, where he only uses the completeness of real numbers, following an idea of Ampère (yes, that Ampère you know) and then linking it to a theorem of Paul Levy.
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function whose partial differentials, increment inequality, increment inequalities, continuous sublinear functional, fundamental kernel, real normed space, mean value inequality, rectifiable path, linear homeomorphism, compact subinterval, monotonicity theorem, continuous linear function, rth derivative, maximal solution, open convex set, converging uniformly, strict local minimum, closed line segment Key Phrases - Capitalized Phrases (CAPs): (learn more)
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function whose partial differentials, increment inequality, increment inequalities, continuous sublinear functional, fundamental kernel, real normed space, mean value inequality, rectifiable path, linear homeomorphism, compact subinterval, monotonicity theorem, continuous linear function, rth derivative, maximal solution, open convex set, converging uniformly, strict local minimum, closed line segment Key Phrases - Capitalized Phrases (CAPs): (learn more)
The Fréchet, Vallée Poussin Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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