Name: Mathematical Analysis
ISBN: 9780201002881

What Hacker News says

jordibcJan 2023

For sure I got significantly(?) better 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:

* Street-Fighting Mathematics, by Mahajan (http://streetfightingmath.com/). Intuitive, useful and fun.

* Geometric Algebra for Physicists, by Doran and Lasenby. I found the power and elegance of geometric algebra mesmerizing, and even if this book is also about physics and there may be more appropriate math-only books about geometric algebra, this is the one that made it for me.

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graycatJan 2019

Part III

Rudin has a nice chapter on Fourier series, that is, representing a periodic function f(t) with a sum of e^(iwt) for overtones at frequencies w and where each frequency w is a whole number multiple of a fundamental frequency. The linear algebra and geometry here are that the e^(iwt) are perpendicular projections of the f(t). So, the e^(iwt) are orthogonal axes.

The line integral generalizes to the exterior calculus of differential forms (keep track of signs from the direction do the integrations, i.e., as in line integrals) and the fundamental theorem of calculus generalizes to Stokes theorem crucial in Maxwell's equations, fluid flow, and partial differential equations. Now we are close to differential geometry, and Rudin also does the inverse and implicit function theorems 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 enough for what physics and engineering do with vector analysis and Stokes theorem.

In H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis, can see the Lebesgue integral, due to H. Lebesgue near 1900. In essentially all cases where the Riemann integral is defined, the Lebesgue integral is also defined and gives the same numerical value. But the Lebesgue integral has more powerful theorems and is defined in more general situations.

As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid foundation to probability, statistics, and stochastic processes.

In W. Rudin, Functional Analysis get a treatment of distributions that cleans up what physics tries to do with the Dirac delta function and also covers spectral theory.

From there, can go for a Ph.D. For that will need mostly (A) pass the qualifying exams and (B) do some original research. The standards are commonly something like "an original contribution to knowledge worthy of publication" and for publication, "new, correct, and significant".

For the qualifying exams, first pick a department, pure/applied math or some math area in engineering, optimization, probability, economics, computer science, etc., get their description of their qualifying exam topics and references, study, and take and pass the exams. For the research, do some.

One suggestion: For the research, pick a real problem and use some math, at least in part new, to get a good solution. Can get "significant" from the importance of the real problem. Can get much of "new" if have the first or better attack on the real problem. Can get "correct" if do the math with careful theorems and proofs. Can publish if pick an appropriate journal. Might get to regard the math as significant based on what it does for the real problem and not just its contribution to pure math.

Even the best US grad schools are hungry for good students. With a good ugrad math major and good work on independent study, should get a good reception at grad schools and at least a tuition scholarship. In that way I got accepted to grad school at math departments at Cornell, Brown, Princeton, and more.

In a sense, independent study is recommended: (A) That is basically what research profs have to do for all their careers. (B) At least at one time the math department at Princeton stated that no courses were offered for preparation for the qualifying exams, students were expected to prepare for the exams on their own, courses were introductions to research by experts in their fields, and students should have some research underway in their first year.

For a little on how to do the original parts of math research, there is the A. Wiles comment:

"Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly it's all illuminated and you can see exactly where you were. Then you go into the next dark room ...."

That's some of how to learn some math, largely with independent study, and maybe to get a Ph.D. But you might want more: You might want your Ph.D. work to get you a good start on a good career in pure/applied math research/applications. For that, look around, pick up what you can in seminars and conferences, and get what your profs can explain to you.

Broad point: Long the best opportunities in applied math, even with some advanced pure math prerequisites, have been in US national security, especially within 100 miles of the Washington Monument.

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graycatJul 2018

To make the study easier and better rounded, just add some material.

Rudin's Principles is really nice, especially in retrospect once understand it, but going in as a student it can seem quite severe. It's precise but not too severe -- he just makes you go a chapter or two before it becomes clear why he is doing what he is doing.

To help, my nutshell view is that mostly he is just trying to develop the Riemann (Stieltjes) integral. His main result is, the Riemann integral exists for continuous functions on compact sets. So, then, he needs to say what a compact set is. Well, the most relevant example is just a closed interval of real numbers such as [a,b]. So, why compact? Because every continuous function on a compact set is uniformly continuous, and that lets us know that the Riemann sums converge. What is compact? Every open cover has a finite subcover, and that lets us get uniform continuity. And, in R^n, a set is compact if and only if it is closed and bounded. So, Rudin needs to talk about closed versus open -- he does that on metric spaces although really he needs it only on R^n.

So, net, he starts with metric spaces and discusses open, closed, and compact. Then he shows that in R^n, compact is the same as closed and bounded. He shows that a continuous function on a compact set is uniformly continuous. Then, presto, he shows that the Riemann (or Riemann-Stieltjes if you wish) sums converge and the Riemann integral exists.

He does some nice work on infinite sequences and series, and the main reason is that he uses those tools to show lots of limits exist, e.g., for sines, cosines, and Fourier series.

There's more of value in Principles, but IMHO I gave you a good start to make the book easier. I wish I'd had been given that outline when I was working through Principles at 1+ hour a page.

But the Lebesgue integral in Royden is the one to take fully seriously.

Make Halmos the second or third text on linear algebra. And then look at some quantum mechanics where they discuss eigenvalues, eigenvectors, Hermitian, unitary, and the spectral decomposition! Right, the Halmos book is baby Hilbert space. Then look at some applied connections, e.g.,

George E.\ Forsythe and Cleve B.\ Moler, {\it Computer Solution of 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 text but not a good first text. So, before Neveu, look quickly, not very seriously, at whatever, including in some introductory statistics texts.

Also, Breiman's Probability is easier to read than Neveu. So, is K. L. Chung's competitive book. And there are others.

There is some more advanced material, e.g.,

Ioannis Karatzas and Steven E.\ Shreve, {\it Brownian Motion and Stochastic Calculus, Second Edition,\/}

Good luck!

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graycatMay 2018

Music:

Get a piano, look up the basics of how to read music, find the keys on the piano, see my post on music theory and the Bach cello piece, get a recording of some relatively simple music you do like, get the sheet music, and note by note learn to play it. After 3-4 such pieces, get an hour of piano instruction and continue on.

Violin: Much the same except need more help at the start. From my music theory post, learn how to tune a violin. Get a good shoulder rest -- the most popular is, IIRC, from Sweden and is excellent. Look at images of violinists and see what rests they are using. Get Ivan Galamian's book on violin. Start in the key of A major and then branch out to E major and D major. Get some good advice on how to hold the violin and the bow; look at pictures of Heifetz, etc. Learn some scales and some simple pieces, get some lessons, and continue.

Math: High school 1st and 2nd year algebra, plane geometry (with proofs), 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 in applications in physics and engineering. There, relax any desire for really careful proofs; really careful proofs with high generality are too hard, and the generality is nearly never even relevant in applications so far. Maybe do the material again if want to do quantum gravity at the center of black holes or some such; otherwise, just stay with what Apostol has. For exterior algebra of differential forms, try hard enough to be successful ignoring that stuff unless you later insist on high end approaches to differential geometry and relativity theory.

Linear algebra, done at least twice and more likely several times. Start with a really easy book that starts with just Gauss elimination for systems of linear equations -- actually a huge fraction of the whole subject builds on just that, and that is close to dirt simple once you see it.

Continue with an intermediate text. I used E. Nearing, student of Artin at Princeton. Nearing was good but had a bit too much, and his appendix on linear programming was curious but otherwise awful -- linear programming can be made dirt simple, mostly just Gauss elimination tweaked a little.

Mostly you want linear algebra over just the real or complex numbers, but nearly all the subject can also be done over any algebraic field -- Nearing does this. Actually, might laugh at linear algebra done over finite fields, but the laughter is not really justified: E.g., algebraic coding theory, e.g., R. Hamming, used finite fields. But if you just stay with the real and complex numbers, likely you will be fine and can go back to Nearing or some such later if wish.

So, concentrate on eigen values and eigen vectors, the standard inner product, orthogonality, the Gram-Schmidt process, orthogonal, unitary, symmetric, and Hermitian matrices. The mountain peak is the polar decomposition and then singular value decomposition, etc. Start to make the connections with convexity and the normal equations in multi-variate statistics, principle components, factor analysis, data compression, etc.

Then, of course, go for P. Halmos, Finite Dimensional Vector Spaces, grand stuff, written as an introduction to Hilbert space theory at the knee of von Neumann. Used in Harvard's Math 55. Commonly given to physics students as their source on Hilbert space for quantum mechanics. Likely save the chapter on multi-linear algebra for later!

For more, get into numerical methods and applications. You can do linear programming, non-linear programming, group representation theory, multi-variate Newton iteration, differential geometry. Do look at W. Fleming, Functions of Several Variables and there the inverse and implicit function theorems and their applications to Lagrange multipliers and the eigenvalues of symmetric or Hermitian matrices. The inverse and implicit function theorems are just local, non-linear versions of what you will see with total clarity at the end of applying Gauss elimination in the linear case.

Physics

Work through a famous text of freshman physics and then one or more of the relatively elementary books on E&M and Maxwell's equations. Don't get stuck: Physics people commonly do math in really obscure ways; mostly they are thinking intuitively; generally you can just set aside after a first reading what they write, lean back, think a little about what they likely really do mean, derive a little, and THEN actually understand. E.g., in changing the coordinates of the gradient of a function, that's not what they are doing! Instead they are getting the gradient of a surface, NOT the function, as the change the coordinates of the surface. They are thinking about the surface, not the function of the surface in rectangular coordinates.

For more than that, you will have to start to specialize. Currently a biggie is a lot in probability theory. There the crown jewels are the classic limit theorems, that is, when faced with a lot of randomness, can make the randomness go away and also say a lot about it.

For modern probability, that is based on the 1900 or so approach to the integral of calculus, the approach due to H. Lebesgue and called measure theory. In the simple cases, it's just the same, gives the same numerical values for, the integral of freshman calculus but otherwise is much more powerful and general. One result of the generality is that it gives, via A. Kolomogorov in 1933, the currently accepted approach to advanced probability, stochastic processes, and statistics.

That's a start.

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kanishkalinuxMar 2018

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.

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graycatMay 2016

There is a film of a way up, over the top lecture in plane geometry by Andrew Gleason with Tom Apostol.

I saw it broadcast late one night on a PBS station and intended for recording for playing in high school classrooms.

Elegant. Powerful. Nothing like what see in common high school plane geometry texts. I got in a few minutes after the start and recognized a lion by his paw. If you liked plane geometry, then you will be in love with this lecture.

I just did 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 algebra but having to study material much like physicists did it 100 years ago.

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