Real Analysis (Classic Version), 4th edition

For graduate-level courses in Real Analysis.

A modern classic

Real Analysis, 4th Edition covers the basic material that every graduate student should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in undergraduate mathematics and familiarity with the material covered in an undergraduate course on the fundamental concepts of analysis. Patrick Fitzpatrick of the University of Maryland - College Park spearheaded this revision of Halsey Royden's classic text.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.

Hallmark features of this title

• Independent, modular chapters give instructors the freedom to arrange the material into a course according that suits their needs. A chart in the text gives the essential dependencies.
• Content is divided into 3 parts: Part 1, Classical theory of functions, including the classical Banach spaces; Part 2, General topology and the theory of general Banach spaces; Part 3, Abstract treatment of measure and integration.
  • Part 1 is a thorough presentation of Lebesgue measure on the real line and the Lebesgue integral for functions of a single variable. Detailed proofs of all major results are now presented in the text. The concept of uniform inte­grability is now prominently placed. An examination of the concept of weak convergence in the L^p spaces, with an applications to the minimization of con­vex functionals, concludes the first part.
  • Part 2 is now a significantly expanded presentation of abstract spaces: metric, topological, Banach, and Hilbert. Foun­dational results for metric spaces (the Baire Category Theorem), for topological spaces (Urysohn's Lemma and the Tychonoff Product Theorem), and for linear spaces (the Hahn-Banach Theorem) are established and employed to create such basic tools for the analysis of linear operators and functionals as the Open Map­ping Theorem, the Uniform Boundedness Principal, Alaoglu's Theorem, and the Krein-Milman Theorem.
  • Part 3 starts with a presentation of the basic theory of general measure spaces and integration over such spaces, in the absence of any topological or algebraic structure. Lebesgue measure on Euclidean space is examined. Product measures are examined, the main result being Fubini's Theorem. Several selected topics are then explored.

New and updated features of this title

• 50% more exercises than the previous edition gives students the practice they need to learn and master the material. The exercises range from those that confirm understanding of fundamental ideas and results to those that offer significant mathematical challenge; many exercises foreshadow future developments.
• Fundamental results, including Egoroff's Theorem and Urysohn's Lemma are now proven in the text.
• The Borel-Cantelli Lemma, Chebychev's Inequality, rapidly Cauchy sequences, and the continuity properties possessed both by measure and the integral are now formally presented in the text along with several other concepts.