Real Variables by Alberto Torchinsky (PDF)

Description

A modern introduction to the theory of real variables and its applications to all areas of analysis and partial differential equations. The book discusses the foundations of analysis, including the theory of integration, the Lebesque and abstract integrals, the Radon-Nikodym Theorem, the Theory of Banach and Hilbert spaces, and a glimpse of Fourier series. All material is presented in a clear and motivational fashion.

User’s Reviews

Editorial Reviews: About the Author Alberto Torchinsky

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I previously wrote the following paragraphs regarding the inadequacy of the index. One commentator has rebuked me (justly) for reviewing an essentially unread book. Well, I’ve consulted the book often since the original review, and apart from the still-inadequate index, admit that I like it. I’m not prepared to review it in detail, but I wanted to modify my previous ranking to more accurately reflect my updated outlook on the text’s usefulness for teaching real variables. Here is the original review, in which I had given the book one star:I have only perused the book in anticipation of an upcoming class, and used it as a supplemental text to Royden’s Real Analysis. The one-star rating I have given at this point applies only to the index, which is abysmal. The index groups each term under the category to which it belongs. For example, when I look for “Radon-Nikodym Theorem,” it is not located in the “R” section. It is instead in “S” under “Signed Measure” as the main heading with “Radon-Nikodym Theorem” as a subheading! Now, such a topic-oriented index may well be useful, but it should be in addition to the standard index, not in place of it! As another example, if one wants to see what Torchinsky has to say about “spectral decomposition,” one must look not in the “S” section but rather the “C” section under “Compact mapping” where “spectral decomposition” exists as a subheading! This flaw is particularly annoying when using the book as a reference text in support of another text. The index is virtually worthless in this case, unless one is willing to scan the index for each topic of interest. Which, by the way, is not that difficult since the index is only 4.5 pages long.Again, my low rating applies only to the index; so far what I’ve seen in the text itself appears commendable. I hope to write a full review at the end of this semester. I can say that no matter how outstanding the text may be, I will never be able to rate above 4 stars due to the inadequacy of the index.

⭐First and foremost as a reference text, this is inadequate. The index is horrible; multiple times throughout the book, Torchinsky will reference a previous result by name, i.e. Egoroff’s Theorem or the Borel-Cantelli Lemma, and you will wonder, ‘where in the book is this result?’ So you go to the index, but it is not in there. That is just my first qualm. The next qualm is that he proves results out of order; he will frequently prove a result and reference a theorem five or more chapters ahead. Moreover his narration is horrible. He proves important results during his prose explanation and never really states what he has proven, but he references it multiple times. He claims to have a ‘new treatment’ of the Radon-Nicodym-Lebesgue Theorem, when in fact it is essentially von Neumann’s approach except longer. I can not stress enough how frustrating this book is.It got two stars for two reasons. The first is that there are a lot of exercises and some have partial solutions in the back. In my opinion that is the best way to learn measure theory. The second, is the general order in which the subjects appear. Although, he proves things out of order, if I was going to write a book on measure theory the table of contents would be similarly ordered.

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