Real Variables with Basic Metric Space Topology (Dover Books on Mathematics) by Prof. Robert B. Ash (PDF)

Description

Designed for a first course in real variables, this text presents the fundamentals for more advanced mathematical work, particularly in the areas of complex variables, measure theory, differential equations, functional analysis, and probability. Geared toward advanced undergraduate and graduate students of mathematics, it is also appropriate for students of engineering, physics, and economics who seek an understanding of real analysis.The author encourages an intuitive approach to problem solving and offers concrete examples, diagrams, and geometric or physical interpretations of results. Detailed solutions to the problems appear within the text, making this volume ideal for independent study. Topics include metric spaces, Euclidean spaces and their basic topological properties, sequences and series of real numbers, continuous functions, differentiation, Riemann-Stieltjes integration, and uniform convergence and applications.

User’s Reviews

Editorial Reviews: About the Author A Professor Emeritus of Mathematics at the University of Illinois, Robert Ash is the author of three other Dover books: Basic Abstract Algebra, Basic Probability Theory, and Complex Variables and Information Theory.

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

⭐Very readable & informative introduction to real analysis.

⭐It’s cheap ,it’s what I need

⭐Unequivocally one of the best treatments of real variables I have come cross. Transitioning into proof-based courses such as real analysis often makes students feel overwhelmed on how to approach proofs but this book is an invaluable help in that regard.

⭐Not an easy read per se’ but contains some good / solid mathematics and connects analysis with topology which for me is interesting – makes math a bit holistic ….

⭐Proofs in this book is more straightforward, very intuitive and readable. Especially good for non-mathematic major students. I think most people who encounter advanced calculus for the first time would get lost in a bunch of axioms and topology properties, but this book makes everything clear.

⭐Great book to study analysis on your own. Has answers to most problems.

⭐Thank you.

⭐I congratulate Prof Ash for his splendid presentation of real variables theory in such a clear manner which is unseen in many books of its kind. Prof Ash seems to know the mentality and anxiety of students taking a course on real variables at this level as he probably faced the same sort of problems when he first took to pure mathematics, having been an engineer by training. What I like about the book is its clarity and the many examples that elucidate the rather hard concepts like uniform convergence. The steps in the proofs are given in great detail which makes reading easy, unlike many other similar books which leave lots of gaps between steps, often leaving the student in a complete state of bewilderment. Nothing is more eirksome to the reader than seeing a phrase like “It is easy to see that blah blah blah” when it is easy to the author who has spent his entire academic life mulling over it but is certainly not easy to the reader who sees it the first time. All writers must understand this. Another good thing about the book is that there are solutions to the exercises that makes self-learning a joy and not a toil.

⭐Probably good for a quick overview or someone who is just entering the topic. I took it up before reading the author’s more thorough “Probability and Measure Theory”. But in hindsight I should have dived straight into the latter.

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