Mathematical Analysis: A Concise Introduction (Hardcover)

Bernd S. W. Schröder



A self-contained introduction to the fundamentals of mathematical analysis

Mathematical Analysis: A Concise Introduction presents the foundations of analysis and illustrates its role in mathematics. By focusing on the essentials, reinforcing learning through exercises, and featuring a unique "learn by doing" approach, the book develops the reader's proof writing skills and establishes fundamental comprehension of analysis that is essential for further exploration of pure and applied mathematics. This book is directly applicable to areas such as differential equations, probability theory, numerical analysis, differential geometry, and functional analysis.

Mathematical Analysis is composed of three parts:

?Part One presents the analysis of functions of one variable, including sequences, continuity, differentiation, Riemann integration, series, and the Lebesgue integral. A detailed explanation of proof writing is provided with specific attention devoted to standard proof techniques. To facilitate an efficient transition to more abstract settings, the results for single variable functions are proved using methods that translate to metric spaces.

?Part Two explores the more abstract counterparts of the concepts outlined earlier in the text. The reader is introduced to the fundamental spaces of analysis, including Lp spaces, and the book successfully details how appropriate definitions of integration, continuity, and differentiation lead to a powerful and widely applicable foundation for further study of applied mathematics. The interrelation between measure theory, topology, and differentiation is then examined in the proof of the Multidimensional Substitution Formula. Further areas of coverage in this section include manifolds, Stokes' Theorem, Hilbert spaces, the convergence of Fourier series, and Riesz' Representation Theorem.

?Part Three provides an overview of the motivations for analysis as well as its applications in various subjects. A special focus on ordinary and partial differential equations presents some theoretical and practical challenges that exist in these areas. Topical coverage includes Navier-Stokes equations and the finite element method.

Mathematical Analysis: A Concise Introduction includes an extensive index and over 900 exercises ranging in level of difficulty, from conceptual questions and adaptations of proofs to proofs with and without hints. These opportunities for reinforcement, along with the overall concise and well-organized treatment of analysis, make this book essential for readers in upper-undergraduate or beginning graduate mathematics courses who would like to build a solid foundation in analysis for further work in all analysis-based branches of mathematics.



- 第一部分介紹了單變量函數的分析,包括數列、連續性、微分、黎曼積分、級數和勒貝格積分。書中詳細解釋了證明寫作,特別關注標準證明技巧。為了方便更抽象的設置,單變量函數的結果使用可轉換為度量空間的方法進行證明。
- 第二部分探討了文本中早期概念的更抽象對應物。讀者將介紹分析的基本空間,包括Lp空間,並成功地詳細介紹了適當的積分、連續性和微分定義,為進一步研究應用數學奠定了強大且廣泛適用的基礎。然後,將檢查測度論、拓撲和微分之間的相互關係,以證明多維替換公式。本部分還涵蓋了流形、斯托克斯定理、希爾伯特空間、傅立葉級數的收斂和黎曼表示定理等內容。
- 第三部分概述了分析的動機以及它在各個學科中的應用。特別關注普通和偏微分方程,介紹了這些領域存在的一些理論和實際挑戰。主題涵蓋了納維爾-斯托克斯方程和有限元方法。