A First Course in Real Analysis (Undergraduate Texts in Mathematics)

Rating:
Summary: Most Complete undergraduate text in Functional Analysis
Review: A must buy for students who want to major in Math or pursue graduate study in Math. The book will prepare a student for higher courses in Functional & Real Analysis

Rating:
Summary: This is one place to begin
Review: First off, this is not a text dedicated to Functional Analysis and, I assume, anyone capable of reading the title would know that. If you are encountering, for the first time, the world of abstract mathematics by way of some class in Real Analysis, then this text is a nice and almost comfortable place to start. This text is not overwhelming terse and lacking in examples like, say Lang's text, and, conversely, this text does not baby you and waste your time with verbose explanations and or proof like, say Strichartz's text. Instead, I would say that this text stands above middle of the road beginning analysis texts and, in general, contains ample exercises as well as examples. Through out the entire text, the authors do an excellent job of not loosing sight of the fact that mathematics is about rigor, about intuitive understanding of abstract theoretical concepts. Remember, this text falls into a class of texts, such as Herstein's, whose purpose (not necessarily primary) is to introduce the reader/student to elegant and more abstract concepts and not to assume that you are inherently gifted with the ability to quickly absorb these new abstract ideas and methods. That is, and with respect to students, this text is for persons with either very limited exposure to analysis or for those who are meeting analysis for the first time. Whereas, the existence of such nice and well thought-out examples make this book a must have as a reference. Especially for those engaged in more abstract higher analysis since the examples that have been given provide an excellent reminder that mathematics is built from the ground up, like a growing pyramid, and highlights the necessary role of abstraction. Finally, this text is thick and covers alot of material. Do not let this fact overwhelm you and in turn persuade you to not engage in reading it. Just take your time and suffer through it like all good mathematicians.