Mathematical Analysis
by Tom M. Apostol · first published 1957
67% recommend33% neutral
Why HN recommends it
Rigorous, clearly written introduction to real analysis that builds proof skills from the ground up; praised for its concise style and intuition, suiting students moving into proof-based math or physicists.
Summarised by Claude from recommending comments
It provides a transition from elementary calculus to advanced courses in real and complex function theory and introduces the reader to some of the abstract thinking that pervades modern analysis.
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Walter Rudin's book on Real Analysis is normally considered as classic in this area. However, you can also take a look at Mathematical Analysis by Tom M. Apostol, which helps in developing good intuition about the subject.
on “Ask HN: Resources to learn real analysis?” ↗…Linear Algebraic Systems,\/} Maybe spend an evening on the documentation of LINPACK. Some weekend of great fun, take a fast pass through the Gauss, ..., Stokes theorem parts of Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced Calculus,\/} where don't take the proofs very seriously but to see how physical science and engineering look at calculus of several variables. Neveu is a great last probability…
on “Ask HN: What are your favorite statistics and probability textbooks?” ↗My recommendation is "One-Variable-Calculus with and introduction to Linear Algebra" by Tom M. Apostol, followed by "Mathematical Analysis". Both books form a great combination!
on “Ask HN: Math books like SICP?” ↗…a Google search looking for a copy on YouTube and found nothing. Anyone know where that lecture is? One of my favorite books, I worked to buy used long after it was out of print, is Tom M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Massachusetts, 1957. And that is advanced calculus more like your ugrad physics prof will see it. Great to have when studying Rudin, exterior…
on “Caltech Professor Emeritus, Author Tom M. Apostol, Dies at 92” ↗Apostol's calculus texts were my first experience with 'real' proof-based mathematics, and I also learned real analysis through his Mathematical Analysis book. Good times; his books taught me a lot. He wrote with clarity and conciseness; when I read Apostol it always seemed like not a single word was added unnecessarily. I highly suggest picking up his…
on “Caltech Professor Emeritus, Author Tom M. Apostol, Dies at 92” ↗…with classics like Spivak, Apostol, Rudin. "Real and Complex Analysis" by Rudin, and the two books both named "Calculus" from Spivak and Apostol. But also from Apostol his more concise and far-reaching "Mathematical Analysis". And from Spivak his small gem "Calculus On Manifolds" made quite a dent on me. Other than more "classic math" books, I also wanted to mention two outliers that I found eye-opening and generally awesome:…
on “Ask HN: Math books that made you significantly better at math?” ↗…of the class in an advanced class in general relativity (e.g., Misner, Thorne, and Wheeler) but will still be lost in much of old physics and engineering! So, what to do? Sure, go to Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced Calculus', Addison-Wesley, Reading, Massachusetts, 1957. The good thing about this book is the lie in the title -- it's mostly a 19th century treatment and not "modern"!…
on “Ask HN: Learn Math the Hard Way” ↗…trigonometry, and hopefully also solid geometry. Standard analytic geometry and calculus of one variable. For calculus of several variables and vector analysis, I strongly recommend Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced Calculus,\/} Addison-Wesley, Reading, Massachusetts, 1957.\ \ Get a used copy -- I did. Actually, it's not "modern" and instead is close to what you will see and need…
on “Ask HN: How do I learn math/physics in my thirties?” ↗…important in differential geometry and parts of nonlinear optimization, e.g., Lagrange multipliers. I warmly suggest that take a really fun, one weekend, pass through Stokes theorem parts of Tom M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Massachusetts, 1957. Get it used, and pay whatever you have to. This way get to see the 2-3 dimensional versions and, really, mostly…
on “Ask HN: How to self-study mathematics from the undergrad through graduate level?” ↗…ignore this, still get correct answers to the homework problems, and sleep fine at night. Which "respectable" book(s) would you recommend for those who want to dive into this details? Is Tom Apostol's Mathematical Analysis? good for learning these kind of details? (They say this book is "respectable", but I would like to hear your thoughts about it. Thanks).
on “Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares” ↗I don't know anything about Apostol's Mathematical Analysis. My guess would be that it demands a fairly sophisticated background of the reader, and does an excellent job of covering calculus from an extremely rigorous point of view. I have heard that Apostol's…
on “Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares” ↗…Vector Spaces, Second Edition,\/} D.\ Van Nostrand Company, Inc., Princeton, New Jersey, 1958.\ \ R.\ Creighton Buck, {\it Advanced Calculus,\/} McGraw-Hill, New York, 1956.\ \ Tom M.\ Apostol, {\it Mathematical Analysis: Second Edition,\/} ISBN 0-201-00288-4, Addison-Wesley, Reading, Massachusetts, 1974.\ \ H.\ L.\ Royden, {\it Real Analysis: Second Edition,\/} Macmillan, New York, 1971.\ \ Walter Rudin, {\it Real…
on “Fermat's Last Theorem – how it’s going” ↗…forms, and the inverse and implicit function theorems. The first half of Rudin's Real and Complex Analysis does well with measure theory and introductions to functional analysis. The Apostol text is Mathematical Analysis: A Modern Approach to Advanced Calculus -- it's NOT very "modern" but gives a lot of what need to know to read mathematical physics, especially Maxwell's equations. For stochastic processes, Doob, Karatsas…
on “Teachers Quit Jobs at Highest Rate on Record” ↗…students complain about how tough the text for our Real Analysis class was, so I was surprised to find it felt pretty readable to me. Turns out it was a new prof that semester who decided to go with "Mathematical Analysis, Second Edition by Tom Apostol" instead of Baby Rudin. Point being, there may be a better analysis text for this student to start with right now- depends highly on their background/situation, but personally…
on “Ask HN: Recommend a maths book for a teenager?” ↗