08-17-2026, 02:31 PM
Real and Abstract Analysis
Authors: Edwin Hewitt & Karl Stromberg
First published: 1965
Publisher: Springer-Verlag
Series:Graduate Texts in Mathematics, Vol. 25
Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable is a classic graduate-level text designed to build a rigorous foundation in modern real analysis while showing how abstract methods grow naturally out of classical analysis. It begins unusually far back, with set theory, relations, the axiom of choice, cardinal and ordinal numbers, and even constructions of the real and complex number systems. It then develops topology and spaces of continuous functions before moving into the book's central subject: measure and integration. The treatment progresses from the Riemann–Stieltjes integral through general measure theory and the Lebesgue integral, emphasizing precise definitions, complete proofs, and general versions of the major theorems.
The later material connects real analysis with functional analysis, introducing normed spaces, Banach and Hilbert spaces and using these ideas in applications such as Fourier analysis and special functions. Despite the word Abstract in the title, abstraction is primarily a tool rather than the objective: the main emphasis remains integration, differentiation, functions, and measure. The authors deliberately present important results both in accessible forms and in greater generality, making the book usable as a graduate textbook, a self-study text, and a reference. It is nevertheless demanding: readers are expected already to possess a solid undergraduate background in rigorous analysis comparable to Apostol's Mathematical Analysis or Rudin's Principles of Mathematical Analysis.
Key takeaways
Goodreads — Real and Abstract Analysis
Authors: Edwin Hewitt & Karl Stromberg
First published: 1965
Publisher: Springer-Verlag
Series:Graduate Texts in Mathematics, Vol. 25
Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable is a classic graduate-level text designed to build a rigorous foundation in modern real analysis while showing how abstract methods grow naturally out of classical analysis. It begins unusually far back, with set theory, relations, the axiom of choice, cardinal and ordinal numbers, and even constructions of the real and complex number systems. It then develops topology and spaces of continuous functions before moving into the book's central subject: measure and integration. The treatment progresses from the Riemann–Stieltjes integral through general measure theory and the Lebesgue integral, emphasizing precise definitions, complete proofs, and general versions of the major theorems.
The later material connects real analysis with functional analysis, introducing normed spaces, Banach and Hilbert spaces and using these ideas in applications such as Fourier analysis and special functions. Despite the word Abstract in the title, abstraction is primarily a tool rather than the objective: the main emphasis remains integration, differentiation, functions, and measure. The authors deliberately present important results both in accessible forms and in greater generality, making the book usable as a graduate textbook, a self-study text, and a reference. It is nevertheless demanding: readers are expected already to possess a solid undergraduate background in rigorous analysis comparable to Apostol's Mathematical Analysis or Rudin's Principles of Mathematical Analysis.
Key takeaways
- Comprehensive foundation: It connects set theory, topology, measure theory, integration, differentiation and functional analysis within one coherent treatment.
- Measure and integration are central: The development of integration from classical ideas to general measure spaces is arguably the heart of the book.
- Rigorous and advanced: Definitions and proofs are given carefully and often at considerable generality, making this better suited to advanced undergraduates or graduate students than beginners.
- Still valuable as a reference: Although originally published in 1965, its treatment of the foundations of analysis remains mathematically relevant; the Mathematical Association of America describes it as a very thorough treatment of classical analysis and recommends it for undergraduate mathematics libraries.
Goodreads — Real and Abstract Analysis
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