Real Analysis (Dover Books on Mathematics) by Edward James McShane (PDF)

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This text offers upper-level undergraduates and graduate students a survey of practical elements of real function theory, general topology, and functional analysis. Beginning with a brief discussion of proof and definition by mathematical induction, it freely uses these notions and techniques. The maximality principle is introduced early but used sparingly; an appendix provides a more thorough treatment. The notion of convergence is stated in basic form and presented initially in a general setting. The Lebesgue-Stieltjes integral is introduced in terms of the ideas of Daniell, measure-theoretic considerations playing only a secondary part. The final chapter, on function spaces and harmonic analysis, is deliberately accelerated. Helpful exercises appear throughout the text. 1959 edition.

User’s Reviews

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

⭐The preface: “this real analysis text accessible to the mature senior-undergraduate, or the beginning graduate.”A senior-undergraduate student might find this text quite challenging. It is assumed (preface) that the student has been exposed to fundamentals already (as assimilated through Hardy: A Course of Pure Mathematics).McShane and Botts, to my knowledge, remains distinct:(1) The preliminary chapter (zero) reminds us: “every function is a relation” and “it is thus more satisfactory to define a function as a graph.” This brief chapter concludes with a definition of factorial function (by induction). Preliminaries will be relatively easy to digest, as it gets much more challenging as we move forward. Also, bold-faced typography accentuates important items throughout the prose (hardcover, first pressing, Van Nostrand).(2) Chapter One, Real Numbers. Read: “an ordered field necessarily has infinitely many elements.” (page 14). As usual, notation is key to unlocking the door to all–as notation for greatest lower bound and least upper bound (page 20). Attention to “the extended-real-number system” will be repaid time and time again throughout (page 26). Attention to section eight will be repaid time and time again: that is partially ordered sets and lattices (page 27).(3) Chapter Two, convergence. Here, it is all about direction: directed sets and directed functions. Exceptional clarity is hallmark of the discussion. Learn terminology: isotone, antitone, monotone (page 58). The big inequalities conclude the chapter: Cauchy-Schwartz, Minkowski. Inequalities, learn of them !(4) Following convergence, continuity. You get the intermediate-value theorem (Darboux property, page 71). You learn that “in metric spaces the basic neighborhoods are spheres.” (page 77). Learn: Lipschitz condition. Then a highlight, as Stone-Weierstrass theorem is approached via introductory probability theory (page 87). Very nice !(5) Next, bounded variation: Thus far we have seen a few novelties (directed sets, for instance), be prepared for more: the derivative will flow from the discussion of bounded variation and “functions on intervals.” Volume of an interval will be introduced (page 108). Mean value theorem, Taylor’s theorem, implicit-function theorem–all here. Another novelty (perhaps) utilizing a dose of linear algebra in the discussion of implicit-function theorem (page 121).(6) We have completed half of the book, now comes Lebesgue-Stieltjes integration. Recall lattices, volume and semi-continuity (page 73), as they will figure prominently in the discussion. Step-Functions first (that is, not a ‘measure theory approach’ first). Thus, less abstract than it would otherwise be ! Directed functions, you get it again (page 128). This is a long chapter (fifty pages). Learn terminology: summable (page 133). A highlight: Fubini’s theorem and its proof: “generalization of the familiar elementary theorem that a double integral can be computed as an iterated integral (single integral of a single integral).” (page 143). If the proof here delights you, all is well !Measure, that comes next (see page 156). A highlight: further discussion of inequalities (pages 161-165). Nice !(7) Continuing with integration: terminology of Boolean rings, sigma-algebras. Some slick proofs, as for Radon-Nikodym theorem (page 181) or integration-by-parts for Lebesgue integral (page 191) or, change-of-variables in Lebesgue integral (page 194). A highlight: Substitution theory for multiple integrals (pages 196-200).(8) Concluding chapter is another long one (fifty pages). An introductory exposition of functional analysis (function spaces, vector space) begins; after which, proving some big theorems (Hahn-Banach, Riesz) on way to Hilbert Space. Twenty-pages of exposition will introduce Hilbert space and Fourier analysis (series and transforms). Nice !(9) Concluding my cursory review: This is one of the more challenging real analysis textbooks (perhaps all are).The prose is terse, yet lucid. The exercises are scattered throughout: they vary widely in difficulty, but, hints supplied.You will not find copious illustrations, figures or diagrams. Much here is analytical (as opposed to geometrical).That which is offered here, in this manner, is difficult to find elsewhere (at an introductory vantage).Note: A copy of Beardon’s Limits, A New Approach to Analysis (1997) is excellent precursor.

⭐I cut my teeth with this book (among others)in learning real analysis starting in the 1960’s. McShane was a major expositor of integration theory in the mid 20th century. Although the book came out in the late 1950’s, it is thoroughly modern and up to date. Most importantly, this book contains valuable material that is difficult to find elsewhere, including an elegant treatment of the Daniell Integral and a detailed proof of the uniqueness of the complete ordered field structure of the real numbers. This Dover edition is a great value and should be in the library of every math enthusiast.

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