# How is "Advanced Calculus" by Loomis/Sternberg?

• Analysis

## Main Question or Discussion Point

Dear Physics Forum friends,

my old friend recently gave me a book called "Advanced Calculus" by Loomis/Sternberg, published by the World Scientific. I had taken a look at that book, and it seems to be that the book treats the vector calculus, linear algebra, and some topics on the mathematical physics. I am currently taking courses that use Rudin's PMA (Chp.1-7) and Hoffman/Kunze's linear algebra texts. I am curious if it will be a productive idea to start reading Loomis/Sternberg after finishing those the single-variable analysis from Rudin and the linear algebra from Hoffman/Kunze. The Loomis/Sternberg book is very tempting to me and I like its conciseness, but I am not sure how should I use it. Is it safe to read Loomis/Sternberg alongside with Rudin's RCA? Do I need to read the multivariate analysis from Rudin first?

Thanks,

## Answers and Replies

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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

mathwonk
Homework Helper
L&S is a very abstract treatment of differential calculus in Banach spaces and on manifolds, and treats integration in finite dimensions a la the theory of "content", which is less advanced than measure theory, and yields less useful theorems of convergence. Rudin's PMA is an austere treatment of some more basic material on constructing real numbers, convergence in metric spaces, and then differential and integral calculus in finite dimensions, not Banach spaces or manifolds I believe, but including elementary measure theory, Then the book RCA repeats measure theory for reals, again in finite dimensions and probably abstract measure spaces, plus complex analysis, including some pretty advanced approximation results.

I would say you could read L&S alongside PMA profitably, but none of those books are recommended for their user friendliness to learners in my opinion, although all are considered authoritative for their content. I took the advanced calculus class from Loomis, taught out of PMA to seniors, and studied out of RCA as a grad student preparing for PhD prelims. I seldom consult any of those books any more, since they just don't offer me much insight. The material is so tightly packaged in those books it is difficult to unpackage it and get it into my brain. Rudin's books did not even make the cut of what to keep on my shelf when I moved. But only you can answer the question of whether they help you, and you should not rule them out based on what I say without vetting them yourself, as many people swear by them. In particular I do not understand your question "is it safe...?" what harm could possibly be done by reading these books?

Look here for other opinions:

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L&S is a very abstract treatment of differential calculus in Banach spaces and on manifolds, and treats integration in finite dimensions a la the theory of "content", which is less advanced than measure theory, and yields less useful theorems of convergence. Rudin's PMA is an austere treatment of some more basic material on constructing real numbers, convergence in metric spaces, and then differential and integral calculus in finite dimensions, not Banach spaces or manifolds I believe, but including elementary measure theory, Then the book RCA repeats measure theory for reals, again in finite dimensions and probably abstract measure spaces, plus complex analysis, including some pretty advanced approximation results.

I would say you could read L&S alongside PMA profitably, but none of those books are recommended for their user friendliness to learners in my opinion, although all are considered authoritative for their content. I took the advanced calculus class from Loomis, taught out of PMA to seniors, and studied out of RCA as a grad student preparing for PhD prelims. I seldom consult any of those books any more, since they just don't offer me much insight. The material is so tightly packaged in those books it is difficult to unpackage it and get it into my brain. Rudin's books did not even make the cut of what to keep on my shelf when I moved. But only you can answer the question of whether they help you, and you should not rule them out based on what I say without vetting them yourself, as many people swear by them. In particular I do not understand your question "is it safe...?" what harm could possibly be done by reading these books?

Look here for other opinions:

Dear Professor Mathwonk,

Thank you very much for your detailed advice! I did not take a deep look into the Loomis/Sternberg, but it seems that the book suits my style as it is very concise and leaving details for me to figure out. I really like Rudin's PMA for the same reason. I can say my mathematical maturity and thinking we're exponentially increased by studying with the Rudin's PMA. After completing my current Analysis I course, I am planning to start studying the multivariate analysis (later chapters on Rudin and Loomis/Sternberg) and Rudin's RCA together starting on the winter. I heard that Rudin's treatment of multivariable analysis is not great, so my plan is to read L/S. If it is comfortable with you, could you recommend me some alternative books for the multivariable-analysis portions of the Rudin and L/S? Also I see that Rudin's RCA is not as popular as a first introduction to the real analysis and complex analysis.....Is there a reason why RCA is not popular as Follnd, Royden, Stein/Sharkachi, and Ahlfors?

mathwonk
Homework Helper
books i recommend include spivak's calculus, apostol's calculus, courant's calculus, spivak's calculus on manifolds, wendell fleming's calculus of several variables, lang's anaklysis I, lang's complex analysis, henri cartan's book on complex analysis, and anything by george simmons or sterling berberian. as stated here often before, rudin's books do not provide any insight or motivation at all into the material presented, just bare proofs.

books i recommend include spivak's calculus, apostol's calculus, courant's calculus, spivak's calculus on manifolds, wendell fleming's calculus of several variables, lang's anaklysis I, lang's complex analysis, henri cartan's book on complex analysis, and anything by george simmons or sterling berberian. as stated here often before, rudin's books do not provide any insight or motivation at all into the material presented, just bare proofs.
Thank you very much for the suggestion! I am going to check them out during this weekend and to see which suits my taste. I have the Calculus, Differential Equations, and Introduction to Topology and Modern Analysis by G. Simmons! I had read his calculus book and the topology sections, and I was very impressed by his clear expositions. But I am not sure if his topology-analysis book covers similar materials as other real analysis books. I have two-volume set of Apostol's Calculus, but I do not like his Calculus II book as I found Hoffman/Kunze and Simmons as better books than Apostol's treatment of linear algebra and differential equations, respectively.

May I ask you for one more advice? What is your criteria of choosing a book for self-studying? How do you decide which one fits you the most?

mathwonk
Homework Helper
i just look at them myself rather than, or after, asking other people. a relevant story: long ago as a grad student i understood my roommate to say that Homological Algebra by Cartan and Eilenberg was a bad book and I avoided it. Then years later at a conference I was required to read some parts of it for a talk I was preparing, and found it extremely clear and precise and instructive. I went back and spoke again to my roommate who said he was merely repeating advice from his brother who when asked said he had only complained it was too precise, or maybe "too meticulous", for his taste. Hence I had avoided one of the best written books of the ages for years for absurd reasons.

i just look at them myself rather than, or after, asking other people. a relevant story: long ago as a grad student i understood my roommate to say that Homological Algebra by Cartan and Eilenberg was a bad book and I avoided it. Then years later at a conference I was required to read some parts of it for a talk I was preparing, and found it extremely clear and precise and instructive. I went back and spoke again to my roommate who said he was merely repeating advice from his brother who when asked said he had only complained it was too precise, or maybe "too meticulous", for his taste. Hence I had avoided one of the best written books of the ages for years for absurd reasons.
I just want to let you know that I found Courant's three-volume sets (currently focusing on the Volume I) and Pugh's Real Mathematical Analysis to be better books than Rudin. As for Courant, I am pleasantly shocked by his brilliant exposition and lots of interesting applications to the mathematical physics, and I found Pugh to be extremely clear and motivating. I also read the first chapter of Serge Lang's Undergraduate Analysis (if that is what you meant by Analysis I), but I did not like his exposition as much as I like his two-volume sets on the calculus and a book on the precalculus....

mathwonk
Homework Helper
I am glad you found some books you like. Unloike most people, Lamg rewrote frequently his books under different titles with slightly varying content. Thje 1968 book Analysis I, was reissued in 1983 as Undergraduate analysis, with some few sections omitted and others added. It now is about 180 pages longer than originally.

mathwonk
Homework Helper
In general Lang's books are poorly organized globally but often read quite well locally. They consist of separately written chapters or sections on different topics, and some sections are very helpful and others are not. There are also sometimes serious errors or misstatements, as well as large omissions, and usually insufficient examples and problems. But there are enough cases where he makes a well phrased explanation of something so clear and brief, that it forever resonates. I.e. one can often learn a single topic or concept extremely well from Lang, but one seldom treasures reading an entire book of his. My assumption is that this was a brilliant man who understood many things well, but who spent little effort planning or polishing his works, just creating them by sticking together individual expositions of varying quality. Lang's books have valuable qualities and many flaws, while in contrast, the calculus books by spivak, courant, and apostol are masterpieces.

In general Lang's books are poorly organized globally but often read quite well locally. They consist of separately written chapters or sections on different topics, and some sections are very helpful and others are not. There are also sometimes serious errors or misstatements, as well as large omissions, and usually insufficient examples and problems. But there are enough cases where he makes a well phrased explanation of something so clear and brief, that it forever resonates. I.e. one can often learn a single topic or concept extremely well from Lang, but one seldom treasures reading an entire book of his. My assumption is that this was a brilliant man who understood many things well, but who spent little effort planning or polishing his works, just creating them by sticking together individual expositions of varying quality. Lang's books have valuable qualities and many flaws, while in contrast, the calculus books by spivak, courant, and apostol are masterpieces.
Dear Professor Mathwonk,

Thank you very much for detailed advice once again. I also agree with you about Lang's books (although I only read his Calculus 1-3 books). His Undergraduate Analysis book contains a lot of interesting materials, particularly about the analysis on the normed function, but I would like to use it as more like a reference book to read sections that are interesting to me.

By the way, I also found "Elementary Real and Complex Analysis" by G. Shilov and "Analysis in Euclidean Space" by K. Hoffman to be clear texts too. There are so many books that I never thought to be a better quality than Rudin.

There are so many books that I never thought to be a better quality than Rudin.
That is because Rudin is actually not really high quality. It's just popular.

That is because Rudin is actually not really high quality. It's just popular.
Is there a reason why Rudin-PMA is very popular among many professors? The books (Pugh, Shilov, and Hoffman) that I am reading alonside are better than Rudin in terms of the exposition, motivation, and problem-sets quality...

Is there a reason why Rudin-PMA is very popular among many professors? The books (Pugh, Shilov, and Hoffman) that I am reading alonside are better than Rudin in terms of the exposition, motivation, and problem-sets quality...
I think there are several reasons why Rudin is so popular
1) Good and difficult problems. Sure, a lot of books have better problems. But Rudin's problems are famous.
2) No mistakes. Comparing with other books, Rudin is virtually without errors.
3) And probably the most important reason: Rudin has been used for a long time already. Now there are a lot more analysis books, but decades ago, there was not so much choice. Professors nowadays often go with the book they've grown up with, which is Rudin.
4) Challenging. A lot of book reviews of Rudin say Rudin is essential because it is a challenge, and mathematicians need that challenge. (Although I'd say the challenge is due to poor exposition rather than an actual good challenge).

mathwonk
Homework Helper
As an example of a clear explanation of something in Lang's Undergraduate analysis, he states there that the definition of the Riemann integral is completely forced on you, just by the area formula for a rectangle, once you agree it should have two properties: it should be monotone, i.e. a smaller function should have a smaller integral, and second it should be additive over decompositions of the interval, i.e. if a < c < b, then the integral from a to b should equal the sum of the integrals from a to c and from c to b. if you think about that forces the definition of the integral in terms of step functions. I was put completely at my ease about the apparently complicated definition of the integral after reading that.

As an example of a clear explanation of something in Lang's Undergraduate analysis, he states there that the definition of the Riemann integral is completely forced on you, just by the area formula for a rectangle, once you agree it should have two properties: it should be monotone, i.e. a smaller function should have a smaller integral, and second it should be additive over decompositions of the interval, i.e. if a < c < b, then the integral from a to b should equal the sum of the integrals from a to c and from c to b. if you think about that forces the definition of the integral in terms of step functions. I was put completely at my ease about the apparently complicated definition of the integral after reading that.
Dear Professor Mathwonk,

Thank you for the explanation. I will definitely read that chapter in Lang. My professor also told me that there are better alternatives for the most of Lang's books, except for the Algebra.

Have you heard anything about Strichartz's The Way of Analysis? One of my graduate students in the research group strongly recommended me that book in conjunction with Rudin. Unfortunately, someone checked the copies from the library shelves and Reserves, so I did not acquire a mean to look through it. From what I observe, it is very lengthy.

I also found that Dover Publication has many excellent books in the science and mathematics! My past impression is that Dover's books are not in a good quality due a cheap price, but my impression is a severely wrong one.

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WWGD
Gold Member
2019 Award
Dear Professor Mathwonk,

Thank you for the explanation. I will definitely read that chapter in Lang. My professor also told me that there are better alternatives for the most of Lang's books, except for the Algebra.

Have you heard anything about Strichartz's The Way of Analysis? One of my graduate students in the research group strongly recommended me that book in conjunction with Rudin. Unfortunately, someone checked the copies from the library shelves and Reserves, so I did not acquire a mean to look through it. From what I observe, it is very lengthy.

I also found that Dover Publication has many excellent books in the science and mathematics! My past impression is that Dover's books are not in a good quality due a cheap price, but my impression is a severely wrong one.
Unfortunately, Dover books _were_ cheaper. Their prices have gone up to a good degree. EDIT: Still,
I would thoroughly recommend the book to an undergraduate, if only because it addresses most of the topics at the foundation of what a math graduate will run into in a standard masters/PHD.

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