Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics)

Rating:
Summary: A Classic "Baby" Analysis Book
Review: This book is a standard undergrad. introduction to Analysis. It provides a nice foundation, making you work at reading proofs and solving problems while getting familiar with the basic concepts -- limsups and infs, basics of continuity, compactness, etc. You would perhaps be better served if this using this book is not your first experience with really doing mathematics, e.g. formal proofs, etc. -- though not Spivak's Calculus on Manifolds or one of J.P. Serre's Arithmetic books, this book is more concise than many. Important theorems such as the Stone Weierstrass are proven in a very clean brief way (this may not lead to the most useful of proof styles -- you may find yourself expending precious time on cleaning up proofs -- "does leaving this step in make me look stupid?" -- and perhaps cutting so much that proofs may look "infelicitous."). I also do not remember this book being strong on Lebesgue theory and don't remember discussion of Littlewood's principles, Radon Nikodym, etc. These, the real substance of Real Analysis, are best seen in Royden or Rudin's Real and Complex book.Moreover, some professors prefer the sigma algebra approach to measures -- the wonderful S. Kakutani, for example, who briefly guest taught the class in which I used this book insisted on reteaching measures using sigma algebras.

Rating:
Summary: A Classic "Baby" Analysis Book
Review: This book is a standard undergrad. introduction to Analysis. It provides a nice foundation, making you work at reading proofs and solving problems while getting familiar with the basic concepts -- limsups and infs, basics of continuity, compactness, etc. You would perhaps be better served if this using this book is not your first experience with really doing mathematics, e.g. formal proofs, etc. -- though not Spivak's Calculus on Manifolds or one of J.P. Serre's Arithmetic books, this book is more concise than many. Important theorems such as the Stone Weierstrass are proven in a very clean brief way (this may not lead to the most useful of proof styles -- you may find yourself expending precious time on cleaning up proofs -- "does leaving this step in make me look stupid?" -- and perhaps cutting so much that proofs may look "infelicitous."). I also do not remember this book being strong on Lebesgue theory and don't remember discussion of Littlewood's principles, Radon Nikodym, etc. These, the real substance of Real Analysis, are best seen in Royden or Rudin's Real and Complex book.Moreover, some professors prefer the sigma algebra approach to measures -- the wonderful S. Kakutani, for example, who briefly guest taught the class in which I used this book insisted on reteaching measures using sigma algebras.

Rating:
Summary: Great analysis...
Review: This book is tough to learn from (because it has almost no motivation), but the text is clearly written and easy to understand.

The proofs are elegant and easy to follow. The construction of the reals using dedikind cuts along the rationals is the only construction I've found in introductory books. Other books I used as suplementary to this (Rosenlicht and Bear) did not have this in their texts.

After learning analysis, I find this book to be an excellent reference for anything that I might have forgotten or just didn't understand the first time around.

Rating:
Summary: Possibly the best (math) textbook I've seen yet
Review: This book served as my introduction to analysis and to higher math itself. It uses strict "French-style" definition-theorem-proof format, and you might find yourself spending hours on one page. Some so-called "maturity" might be needed in order to submit to what look like unrelated strands of thought in order to reach desired results. But any bright student will be able to handle the material. Let me emphasize this: NO prior introduction to mathematics of ANY sort is NEEDED for reading this book. Nothing besides a logical, patient mind is a prerequisite. Although tastes may vary, I have a revulsion towards math books which do not use this format, but go for a looser, more conversational style. I believe they are much less clear and more difficult for studying introductory material. I suggest that you use this book as your primary introduction to analysis, and look into other "looser" ones whenever you are having trouble. This book is also excellent reference. If you are trying to learn analysis, and ESPECIALLY IF THIS IS YOUR FIRST COURSE IN HIGHER MATH (!), for God's sake don't get a looser book. You will have made a mistake.