Principles of Real Analysis

Again, thirty or forty years ago you wouldn't have found a book like this one for the subject of Real Analysis. First off the authors offer an answer book for the problems given in the text. In the "old days" this just wasn't done. Something about: "you have to suffer to obtain glory" was what one of my Math professors often said. Oh yeah it was:" there is no royal road to learning mathematics". That was his line. Well things change. And now these authors for what ever reason have tried to give us poor "challenged types" a break. Good for them! Usually the complaints about the subject of Real Analysis and its teaching are of a particular or specific type. Like; "I don't get it!". That's usually the nature of the "whining and crying". Probably the crime originates not in University courses so much as it originates in High School Mathematics teaching. That is the High School Mathematics teacher who takes his/her frustrations out on the class. That is where the student first hears the word, "Real Number". "Yeah", they often say "...whad'ya mean by Real? It's a number ain't it? It's gotta be real!" end of quote. I never ever had a Math teacher in High School prove the irrationality of the square root of two. First off that's a proof by contradiction. Yet the Ancient Greeks showed it to be so 2500 years ago. But somehow it never got done in my High School when I was there. Also "Countability". Never got done in my High School. So that by the time you get to University and they throw Cantor's proof for the non-denumerability of the Real numbers at you, you often feel a weird sensation of having been "defiled", somehow "violated". Violated because you really were never taught how to count 1,2,3,...properly.And since an irrational number like the root of two is uncountable you can't go "1, 2, 3,...." pointing at it with your finger. Since you can't point at it with your finger it " really doesn't really exist " as we human beings "know" existence. We know it exists but we can't "point to it". That was what they couldn't get across in High School or even University. What they tried to teach us in High School Algebra of course was factoring. Okay. But then they couldn't teach us the Binomial Theorem because they, the teachers didn't know it themselves. There's that word "know". And then if you did show some desire to learn more they got angry and said: " ...this is way, way above your head!". Whose head? There's or mine? Now I study Real analysis on my own. Certainly not for any monetary profit. I'm what you call an "amateur"...for the "love". This book is an invaluable aid to that end. Best Regards Southern Jameson West

Finished reading those undergraduate analysis books that made a study of metric spaces look like a tall order? Well then reading this book would be an excellent continuation of the hard work. The book is largely about the Lebesgue theory of integration, but includes a very thorough coverage of the theory of metric and topological spaces in the first two chapters. Chapters 3,4 and 5 are the heart of the book covering measure theory, the Lebesgue integral and some topics from introductory functional analysis like theory of operators and Banach spaces. Chapters 6 and 7, covering Hibert spaces, the Radon Nikodym theorem and the Riesz Representation Theorem among other things, are the most useful for someone like me who wants to master higher analysis in order to read financial mathematics. And what's more, there is a solutions book providing answers to all 609 problems (spread over 7 chapters!) and more. All in all, the authors have made a great contribution!

An ideal text for a first-year graduate students in mathematics studying Real Analysis. The exposition is complete and very clear, including a lot of optional material for the curious. A detailed introduction to Functional Analysis is also included. Those needing the infamous Radon-Nikodym theorem and theory of signed measures will need to skip around since this is presented in the very last chapter (not a big problem). Also, consult the authors' companion text Problems in Real Analysis, which could be very useful to those preparing for a qualifying exam in analysis at the PhD level. Overall, a highly recommend text.