Mathematical Analysis (PDF)

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.