The Theory of Functions of Real Variables: Second Edition (Dover Books on Mathematics) 2nd Edition by Lawrence M Graves (PDF)

Description

In scope and choice of subject matter, declared the Bulletin of the American Mathematics Society, “this text is nicely calculated to suit the needs of introductory classes in real variable theory.” A balanced treatment, it covers all of the fundamentals, from the real number system and point sets to set theory and metric spaces.Starting with a brief exposition of the ideas and methods of deductive logic, the text proceeds to the postulates of Peano for the natural numbers and outlines a method for constructing the real number system. Subsequent chapters explore functions and their limits, the properties of continuous functions, fundamental theorems on differentiation, the Riemann integral, and uniform convergence. Additional topics include ordinary differential equations, the Lebesgue and Stieltjes integrals, and transfinite numbers. Useful, well-chosen lists of references to the literature conclude each chapter.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A great book on Real Analysis at the beginning advanced undergraduate/beginning graduate level. A great deal of material on the Lebesgue Integral which provide a thorough development. Very few exercises in the book to help the reader in understanding the material and applying the theory. I learn best by solving problem and proving theorems. Though, this is an excellent book to study the proofs and fill in the missing details.The book covers:Set theory, real number system, point set topology, functions, limits, differentiation, Riemann integral, uniform convergence, Lebesgue integral, Stieltjes integral, and metric spaces.Metric spaces are not introduced until the last chapter; nothing on Banach spaces at all. Of course, this book was written in the 1950’s immediately after modern analysis was developed.

⭐This Dover book, ”

⭐, by Lawrence Murray Graves, is packed with valuable real-number analysis, including deeper analysis of some topics than you will find in modern books. This 1946 vintage book uses logic notations which were fashionable from the time of Peano up until the 1950s, but now they are difficult to read.The detailed 37-page coverage of Stieltjes integrals includes three kinds of integrals, of type S, GS and LS. However, this is beyond what modern books will tell you about Stieltjes integrals, largely because the Lebesgue integral is now dominant. Graves gives 87 pages on the Lebesgue integral, mostly using the Riesz linear functional approach, as opposed to the measure theory approach where integrals are built up from measures of sets.This book includes a fair amount of material on logic, set theory, the axiomatic approach to the positive integers (including the Peano axioms), the Dedekind cut representation of the real numbers, the Bolzano-Weierstrass and Heine-Borel theorems, plus also a 38-page chapter on fairly naive set theory.For people who want to understand the historical progression of real analysis in the 20th century, this book is valuable in showing the way in which people were thinking in the 1940s. However, the modern emphasis has moved away from many of the nitty-gritty detailed theorems in this book, and notations have moved on too. Anyone who has a hard-core fascination with the history of mathematical analysis will find this book of great value, but for the modern student, it’s probably an inefficient way to soak up the basics of real analysis. The reading level is around about 3rd or 4th year university pure mathematics, I would say.Graves’ approach to the definitions of integers, rational numbers and real numbers is different to the approach taken in most modern books, and maybe his strategy is better. He defines first the positive integers, then the positive rational numbers, and then the positive real numbers using Dedekind cuts. And finally he includes the negative numbers. Anyone who has written out the full technicalities of the definitions of signed numbers will know that much of the hard work comes from the negative number arithmetic implementation details. Graves avoids this by doing everything for the positive numbers first. I think this is a valuable part of the book, pages 17-39.

⭐No problem. Exactly what I needed, and delivered in an incredibly short time

⭐

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