From Classical to Modern Analysis

This innovative textbook bridges the gap between undergraduate analysis and graduate measure theory by guiding students from the classical foundations of analysis to more modern topics like metric spaces and Lebesgue integration. Designed for a two-semester introduction to real analysis, the text gives special attention to metric spaces and topology to familiarize students with the level of abstraction and mathematical rigor needed for graduate study in real analysis. Fitting in between analysis textbooks that are too formal or too casual, From Classical to Modern Analysis is a comprehensive, yet straightforward, resource for studying real analysis.
To build the foundational elements of real analysis, the first seven chapters cover number systems, convergence of sequences and series, as well as more advanced topics like superior and inferior limits, convergence of functions, and metric spaces. Chapters 8 through 12 explore topology in and continuityon metric spaces and introduce the Lebesgue integrals. The last chapters are largely independent and discuss various applications of the Lebesgue integral.
Instructors who want to demonstrate the uses of measure theory and explore its advanced applications with their undergraduate students will find this textbook an invaluable resource. Advanced single-variable calculus and a familiarity with reading and writing mathematical proofs are all readers will need to follow the text. Graduate students can also use this self-contained and comprehensive introduction to real analysis for self-study and review.

Guides undergraduate students from calculus to measure theory and the Lebesgue integral Provides a self-contained presentation of metric spaces and their topology tailored for first-time students of real analysis Includes cumulative exercises that prepare students for real analysis's many applications

Autorentext

Rinaldo Schinazi is a Professor of Mathematics at the University of Colorado, USA.

Zusammenfassung
"This textbook is designed for a two-semester introductory course on real analysis, and its unique feature is that it focuses on both elementary and advanced topics. ... the book is written in an accessible and easy to follow style." (Antonín Slavík, zbMATH 1408.26001, 2019)

Inhalt
Preface.- Real Numbers.- Sequences of Real Numbers.- Limits Superior and Inferior of a Sequence.- Numerical Series.- Convergence of Functions.- Power Series.- Metric Spaces.- Topology in a Metric Space.- Continuity on Metric Spaces.- Measurable Sets and Measurable Functions.- Measures.- The Lebesgue Integral.- Integrals with Respect to Counting Measures.- Riemann and Lebesgue Integrals.- Modes of Convergance.- References.