Charles C. Pugh is Professor Emeritus at the University of California, Berkeley. His research interests include geometry and topology, dynamical systems, and normal hyperbolicity.
Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.
New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri´s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali´s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.
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Elucidates abstract concepts and salient points in proofs with over 150 detailed illustrations
Treats the rigorous foundations of both single and multivariable Calculus
Gives an intuitive presentation of Lebesgue integration using the undergraph approach of Burkill
Includes over 500 exercises that are interesting and thought-provoking, not merely routine
Real Numbers.- A Taste of Topology.- Functions of a Real Variable.- Function Spaces.- Multivariable Calculus.- Lebesgue Theory.
Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri's Principle, and, in many cases, the ability to see an integral result from measure theory.
Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.
New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri's Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali's Covering Lemma, density points - which are rarely treated in books at this level - and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.
"This book, in its second edition, provides the basic concepts of real analysis. ... I strongly recommend it to everyone who wishes to study real mathematical analysis." (Catalin Barbu, zbMATH 1329.26003, 2016)
Über den Autor
Charles C. Pugh is Professor Emeritus at the University of California, Berkeley. His research interests include geometry and topology, dynamical systems, and normal hyperbolicity.
Inhaltsverzeichnis
Real Numbers.- A Taste of Topology.- Functions of a Real Variable.- Function Spaces.- Multivariable Calculus.- Lebesgue Theory.
Klappentext
Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.nnNew to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri¿s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali¿s Covering Lemma, density points ¿ which are rarely treated in books at this level ¿ and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.
Elucidates abstract concepts and salient points in proofs with over 150 detailed illustrations
Treats the rigorous foundations of both single and multivariable Calculus
Gives an intuitive presentation of Lebesgue integration using the undergraph approach of Burkill
Includes over 500 exercises that are interesting and thought-provoking, not merely routine
