„This highly original, interesting and very useful book includes over 900 exercises which are ranging in levels of difficulty, from conceptual questions and adaptations of proofs to proofs with and without hints.“ (Mathematical Reviews, 2008h)
A self-contained introduction to the fundamentals of mathematicalanalysis
Mathematical Analysis: A Concise Introduction presents thefoundations of analysis and illustrates its role in mathematics. Byfocusing on the essentials, reinforcing learning through exercises, and featuring a unique „learn by doing“ approach, the book developsthe reader's proof writing skills and establishes fundamentalcomprehension of analysis that is essential for further explorationof pure and applied mathematics. This book is directly applicableto areas such as differential equations, probability theory, numerical analysis, differential geometry, and functionalanalysis.
Mathematical Analysis is composed of three parts:
? Part One presents the analysis of functions of one variable, including sequences, continuity, differentiation, Riemannintegration, series, and the Lebesgue integral. A detailedexplanation of proof writing is provided with specific attentiondevoted to standard proof techniques. To facilitate an efficienttransition to more abstract settings, the results for singlevariable functions are proved using methods that translate tometric spaces.
? Part Two explores the more abstract counterparts of the conceptsoutlined earlier in the text. The reader is introduced to thefundamental spaces of analysis, including Lp spaces, and the booksuccessfully details how appropriate definitions of integration, continuity, and differentiation lead to a powerful and widelyapplicable foundation for further study of applied mathematics. Theinterrelation between measure theory, topology, and differentiationis then examined in the proof of the Multidimensional SubstitutionFormula. Further areas of coverage in this section includemanifolds, Stokes' Theorem, Hilbert spaces, the convergence ofFourier series, and Riesz' Representation Theorem.
? Part Three provides an overview of the motivations for analysis aswell as its applications in various subjects. A special focus onordinary and partial differential equations presents sometheoretical and practical challenges that exist in these areas. Topical coverage includes Navier-Stokes equations and the finiteelement method.
Mathematical Analysis: A Concise Introduction includes an extensiveindex and over 900 exercises ranging in level of difficulty, fromconceptual questions and adaptations of proofs to proofs with andwithout hints. These opportunities for reinforcement, along withthe overall concise and well-organized treatment of analysis, makethis book essential for readers in upper-undergraduate or beginninggraduate mathematics courses who would like to build a solidfoundation in analysis for further work in all analysis-basedbranches of mathematics.
Mathematical Analysis: A Concise Introduction presents thefoundations of analysis and illustrates its role in mathematics. Byfocusing on the essentials, reinforcing learning through exercises, and featuring a unique „learn by doing“ approach, the book developsthe reader's proof writing skills and establishes fundamentalcomprehension of analysis that is essential for further explorationof pure and applied mathematics. This book is directly applicableto areas such as differential equations, probability theory, numerical analysis, differential geometry, and functionalanalysis.
Mathematical Analysis is composed of three parts:
? Part One presents the analysis of functions of one variable, including sequences, continuity, differentiation, Riemannintegration, series, and the Lebesgue integral. A detailedexplanation of proof writing is provided with specific attentiondevoted to standard proof techniques. To facilitate an efficienttransition to more abstract settings, the results for singlevariable functions are proved using methods that translate tometric spaces.
? Part Two explores the more abstract counterparts of the conceptsoutlined earlier in the text. The reader is introduced to thefundamental spaces of analysis, including Lp spaces, and the booksuccessfully details how appropriate definitions of integration, continuity, and differentiation lead to a powerful and widelyapplicable foundation for further study of applied mathematics. Theinterrelation between measure theory, topology, and differentiationis then examined in the proof of the Multidimensional SubstitutionFormula. Further areas of coverage in this section includemanifolds, Stokes' Theorem, Hilbert spaces, the convergence ofFourier series, and Riesz' Representation Theorem.
? Part Three provides an overview of the motivations for analysis aswell as its applications in various subjects. A special focus onordinary and partial differential equations presents sometheoretical and practical challenges that exist in these areas. Topical coverage includes Navier-Stokes equations and the finiteelement method.
Mathematical Analysis: A Concise Introduction includes an extensiveindex and over 900 exercises ranging in level of difficulty, fromconceptual questions and adaptations of proofs to proofs with andwithout hints. These opportunities for reinforcement, along withthe overall concise and well-organized treatment of analysis, makethis book essential for readers in upper-undergraduate or beginninggraduate mathematics courses who would like to build a solidfoundation in analysis for further work in all analysis-basedbranches of mathematics.
