Slashdot Mirror

There's a growing trend in math (and maybe other disciplines, for all I know) away from non-free publishing.

Prominent mathematicians have been complaining for years (more links here) about overpriced journals, and entire editorial boards of some journals have resigned in protest (see a list of mass resignations and similar changes here). There are now plenty of entirely free journals in combinatorics, topology, and other fields, and pretty much everything that gets published these days is either available on the author's website or on the arXiv.

So modern research tends to be free, but what about all the books you need to read before you understand this research? Sure, a copy of Rudin may be expensive and there's not much we can do about that, but maybe you can learn from the free analysis course notes at MIT's OCW site. You complain that EGA is out of print, but basically everything Grothendieck wrote is available for free, and you can even get them along with tons of other old French publications through NUMDAM. (There's even a project to transcribe SGA into LaTeX.) Lots of other books are free to download legally (and this is by no means a complete list), even though many are commercially published as well.

Finally, you can complain all you want about university tuition, but I really doubt that free tuition is going to open up mathematics to the masses. Ultimately the very top students who can't afford it are getting scholarships and grants to cover their education (and I do know some people who got free rides at Princeton because they couldn't afford it -- that school is definitely more generous than most), and since most other people couldn't get into Princeton anyway the tuition is never even an issue for them. The best way to make mathematics more accessible is to give everyone access to free textbooks and current research, and the "marxist university professors" you deride have been gradually moving in that direction for years now.

By the way, what do you think has been done to damage the Princeton math department's reputation? Whatever you think Shapiro and Tilghman have done to the university, nobody in their right mind would deny that it's one of the top few in the world and I doubt most people would openly proclaim any one department to be the best anyway.

There's a growing trend in math (and maybe other disciplines, for all I know) away from non-free publishing.

Prominent mathematicians have been complaining for years (more links here) about overpriced journals, and entire editorial boards of some journals have resigned in protest (see a list of mass resignations and similar changes here). There are now plenty of entirely free journals in combinatorics, topology, and other fields, and pretty much everything that gets published these days is either available on the author's website or on the arXiv.

So modern research tends to be free, but what about all the books you need to read before you understand this research? Sure, a copy of Rudin may be expensive and there's not much we can do about that, but maybe you can learn from the free analysis course notes at MIT's OCW site. You complain that EGA is out of print, but basically everything Grothendieck wrote is available for free, and you can even get them along with tons of other old French publications through NUMDAM. (There's even a project to transcribe SGA into LaTeX.) Lots of other books are free to download legally (and this is by no means a complete list), even though many are commercially published as well.

Finally, you can complain all you want about university tuition, but I really doubt that free tuition is going to open up mathematics to the masses. Ultimately the very top students who can't afford it are getting scholarships and grants to cover their education (and I do know some people who got free rides at Princeton because they couldn't afford it -- that school is definitely more generous than most), and since most other people couldn't get into Princeton anyway the tuition is never even an issue for them. The best way to make mathematics more accessible is to give everyone access to free textbooks and current research, and the "marxist university professors" you deride have been gradually moving in that direction for years now.

By the way, what do you think has been done to damage the Princeton math department's reputation? Whatever you think Shapiro and Tilghman have done to the university, nobody in their right mind would deny that it's one of the top few in the world and I doubt most people would openly proclaim any one department to be the best anyway.

Don't forget journals.

... try to read a paper from their journal (JAMS http://www.ams.org/jams/2003-16-03/S0894-0347-03-00422-3/S0894-0347-03-00422-3.pdf) and you will be asked for... money. Well that's their interpretation of "... In mathematics information is passed on free of charge..." cheers

Sorry, I should have linked to this page on the AMS site, which links to many more examples:
http://www.ams.org/STIX/private/stixprv-index.html

Drill down to the Project page: http://www.stixfonts.org/project.html And the American Mathematical Society STIX project page has some examples: http://www.ams.org/STIX/private/stixprv-E2.html

There's a relevant article in this month's Notices:

http://www.ams.org/notices/200710/tx071001279p.pdf

Specifically, the quote from the horse's mouth:

http://reference.wolfram.com/mathematica/tutorial/WhyYouDoNotUsuallyNeedToKnowAboutInternals.html

which contradicts the principles of mathematics, and of scientific method. I can answer *any* question you ask!

For some perspectives on the complete nonprofundity and borderline academic dishonesty of Wolfram's book from some people who _do_ know what they're talking about, see this review (PDF) from the Notices of the American Mathematical Society and this collection of many more links to reviews.

Then I'm afraid you didn't study advanced maths very well. Analysis is an important part of the field, to be sure, but it's no more fundamental than algebra, geometry, logic, and numerous other areas. The American Mathematical Society has a comprehensive classification of these subfields, and it's vast.

The nonsense is free online. Wow, now millions of people can read it, waste time ...and make fun it.. hopefully. Crazy NKS "goodness" for your reading "pleasure": here .

Trust me, even if it is free, after reading it, you'll want your "free" back.

You didn't actually read the damn thing, did you? I'm getting really tired of this mindless NKS bashing, no matter how fashionable it is. A book that was largely favorably reviewed in Notices of the American Mathematical Society cannot be 100% nonsense, can it really? I find it amusing that those who are most critical of NKS are almost never real scientists.

There are some severe flaws with NKS. The fundamental philosophical claims are highly doubtful, the "new science" mentioned in its title does not live to its name, the egomaniacal tone, the passing off of other people's hard work as Wolfram's own, the revisionist history, etc. But that said, there is a lot to enjoy in the book. The footnotes are worth the price of a copy on their own, as they are in many ways one of the best exposés of the history of the 20th century focusing on computer science, mathematics and physics I have ever read.

I knew a lot about CAs and discrete models before reading the book, most likely more than you know, or will ever know, and yet I really did learn a lot from it. You just have to be intelligent and well-versed enough to be able to separate the wheat from the chaff. Maybe that's your real problem with the book?

We agree neither to submit to, referee for, nor participate in the operation of any journal that charges an excessively high per page subscription fee, as compared to the average of the 25 highest impact journals in pure mathematics**
 

Yeah, the shine's definitely gone off Google, eh? at the rate google (and yahoo) are swallowing up other sites there's going to be some major monopolising going on.

I think searching the web is one of the few bastions where closed source still rules, and it surprises me that no-one's really made an open source search engine. I'm aware that there are things like Nutch and ht:dig out there but their scope is completely different (site-wide searching primarily).

So - why don't we have an open source search engine? Pagerank is fairly easy to implement, and would serve as a good starting point for improvement. Writing apps to rank and sort web pages strikes me as the type of problem that a lot of smart people would find a lot of fun.

I know that it requires a crap load of infrastructure, but if Wikipedia can handle it. Besides, you can index one hell of a lot of pages with the standard few GB of bandwidth a month on cheap-ish hosting plans.

So - why not?

The CS degress (typically out of the Math departments) are just crap, and full of useless theory; find a good program that is out of the Engineering department (with no relation to the Math department) and you should be good.

The publishers' real problem is not free journals, but rather libraries trying to stop paying the publishers' vastly increased charges. "Free journals" is merely the latest library tactic.

Many librarians and researchers didn't start out caring about the principle of free journals. The publishers' greed forced it on them. Even major research libraries are cancelling subscriptions. The libraries are doing that because the journal prices have been increasing so fast. First the libraries tried switching from paper copies to online subscriptions. So the publishers raised the online prices. Further, the publishers bundled many journals together so that libraries could not cancel the least used titles. The libraries try to form consortia to share subscriptions, but the publishers' license terms stop that.

Even "free" journals cost someone money. PLOS is quite expensive to publish in. Their model is charging the author not the reader.

One factor driving rising journal prices is the increased concentration as big publishers like Elsevier buy their competitors. Some years ago, Elsevier stated their business model as, approximately, serving assistant profs trying to get tenure. In 2005, their profit was 655 million euros on income of 2097 million euros ( http://www.reed-elsevier.com/ ) That's not a bad profit margin.

Journals are priced like drugs, at what the market is perceived to bear. That can be up to $2/page (w/o even any color) ( http://www.ams.org/membership/journal-survey.html )

Journals are obsolete. They're slow to publish, rarely have color, don't have videos, etc. We academics publish in them because administrators use them to judge us. However, when we need something, we search the web, not the libraries. I put my own research first on the web, so that people can find it. Later I write papers.

Finally, to respond to the comment that publicly funded work should be free: That would be nice, but there's a US law giving universities ownership in discoveries resulting from NSF-funded research. What do other countries do?

Yeah, but can I run Linux on that one? :)

Today you don't have to sit around filing triangular gears. It's been automated.
http://www.ams.org/featurecolumn/archive/kyth5.htm l

And here's the specs http://www.ams.org/featurecolumn/archive/kyth2.htm l

Those old guys did a pretty good job. Without even electric grinding tools.

Electricity! When we wanted electricity, we had to rub amber. Or get a torpedo fish.

Today you don't have to sit around filing triangular gears. It's been automated.
http://www.ams.org/featurecolumn/archive/kyth5.htm l

And here's the specs http://www.ams.org/featurecolumn/archive/kyth2.htm l

Those old guys did a pretty good job. Without even electric grinding tools.

Electricity! When we wanted electricity, we had to rub amber. Or get a torpedo fish.

Yet another slashvertisement for New Scientist claptrap. Will the pseudo science crap ever stop? If I wanted to read that shit I'd go there, PLEASE stop posting it here.

"New" Scientist? If this is the new science I don't want anything to do with it.

At least they do not claim to be scientists, just "New Scientists". New Scientist = euphemism for Pseudo Scientist.

Give us some real science please. You won't find it at New Scientist, nor will you find it in Nature.

You can find real science in publications like those overseen by the following organisations: ACS, RSC, AIP, IOP, AMS, Elsevier, etc., etc...

See the difference? Probably not...

Um, I'm not sure what your comment had to do with the article.

Also, you generally cannot download scientific/mathematical papers free of charge. Yes, the papers on arxiv.org are free, but most journal published papers are not, and in fact subscription to mathematical/scientific/technical journals tend to be very expensive. To put this in perspective, an electronic subscription to the 2007 AMS journal will set you back $155 (link).

E.g., my professor used network algorithms when writing the software that matches medical schools and students. Each school prioritizes it's candidate students, each student prioritizes his schools. The program finds the best matches.

For an interesting mathematical discussion of this problem, see the American Mathematical Society article, "Mathematical Marriages", and scroll down to section 5 ("Hospitals and Residents"). By coincidence, I just happened to run across that article today...

Perhaps it would be more productive to consider awards in math and science for people who do an excellent job of popularising or explaining existing material

"Sometimes it seems that the only really new ideas being tossed around (outside of lab research and the like) in science are from Wolfram in his book, A New Kind of Science."

Really new? No. Tossed around? Oh yes ;-)

" In ANKS Wolfram says that "the core of this book can be viewed as introducing a major gener- alization of mathematics" (p. 7). In this he is entirely mistaken, but there are at least two ways in which he has benefited mathematics: he has helped to popularize a relatively little-known mathematical area (CA theory), and he has unwittingly provided several highly instructive examples of the pitfalls of trying to dispense with mathematical rigor."

I haven't looked at all the posts, but just in case here are some links that might be useful in your search.

http://www.ams.org/employment/
http://www.ams.org/early-careers/
http://www.maa.org/careers/index.html
http://www.ams.org/careers/
http://math.ucsd.edu/~sbuss/GradInfo/index.html
http://www.beanactuary.org/
http://www.nsa.gov/careers/index.cfm
http://www.census.gov/hrd/www/jobs/emp_opp.html

I haven't looked at all the posts, but just in case here are some links that might be useful in your search.

http://www.ams.org/employment/
http://www.ams.org/early-careers/
http://www.maa.org/careers/index.html
http://www.ams.org/careers/
http://math.ucsd.edu/~sbuss/GradInfo/index.html
http://www.beanactuary.org/
http://www.nsa.gov/careers/index.cfm
http://www.census.gov/hrd/www/jobs/emp_opp.html

I haven't looked at all the posts, but just in case here are some links that might be useful in your search.

http://www.ams.org/employment/
http://www.ams.org/early-careers/
http://www.maa.org/careers/index.html
http://www.ams.org/careers/
http://math.ucsd.edu/~sbuss/GradInfo/index.html
http://www.beanactuary.org/
http://www.nsa.gov/careers/index.cfm
http://www.census.gov/hrd/www/jobs/emp_opp.html

Actually for typesetting math with LaTeX (but not needing to be a total complete wizard) you can read: Short Math Guide To LaTeX. I started using LaTex for all my university papers based soley on it!

It's worth mentioning that Carleson was on the faculty at UCLA, usually spending at least the winter quarter there (it doesn't take a genius to prefer Los Angeles in February to Uppsala in February.) I think all of the graduate students he advised were in Sweden though, which seems to be the case from the math genealogy site: http://www.genealogy.ams.org/html/id.phtml?id=1978 1 He did at least intermittently teach the first-year graduate analysis course at UCLA, and made those students suffer (and learn...)

This is indeed common. The same sentiment can be found in an opinion piece (free registration required) published in the August 2005 American Mathematical Society Notices written by Dr. Richard Schaar. Also seen in this slashdot comment.

Oh, you posted that comment. Well, nevermind

I wish it would be as useful as http://www.ams.org/mathscinet MathSciNet is for math papers. I can't find any math paper as easily on google scholar as I can on MathSciNet.

I think they might prove usable. Judge for yourself.

What about the golden ratio (sqrt(5)+1)/2, which is the most irrational number? It gets that name because continued fraction expansions are used to find the best rational approximation to a number (such as approximating pi by 22/7), and it can be proven that the golden ratio has the slowest converging continued fraction approximation.

Where in the world are you getting those numbers from? According to the American Mathematical Society's MathSciReview:

Facts and Figures: 22506 items added in 2005; 1799 journals covered; links to 500526 original articles; 11304 active reviewers; 428094 authors indexed.

Also, I wager you are underestimating the number of papers in error. But even if you weren't, automated proof checking can achieve a lower error rate than 0.1%.

It may become common to publish computer-checkable proofs with normal papers, but that will not happen until creating the computer-checkable proof does not take the mathematician significant extra time. Which will require a computer program that is a lot closer to genuine artificial intelligence than anything we have yet.

This is absolutely correct. Still one has to consider the "extra time" of the reader. If you are right about error rates, then yes, it probably wouldn't make much sense to proof check until the time consumed is significantly shrunk.

Where in the world are you getting those numbers from? According to the American Mathematical Society's MathSciReview:

Facts and Figures: 22506 items added in 2005; 1799 journals covered; links to 500526 original articles; 11304 active reviewers; 428094 authors indexed.

Also, I wager you are underestimating the number of papers in error. But even if you weren't, automated proof checking can achieve a lower error rate than 0.1%.

It may become common to publish computer-checkable proofs with normal papers, but that will not happen until creating the computer-checkable proof does not take the mathematician significant extra time. Which will require a computer program that is a lot closer to genuine artificial intelligence than anything we have yet.

This is absolutely correct. Still one has to consider the "extra time" of the reader. If you are right about error rates, then yes, it probably wouldn't make much sense to proof check until the time consumed is significantly shrunk.

Some of this seems kind of pointless. Since you are posting AC, you might be some physics grad student at Berkeley trying (unsuccessfully) to get a PhD advisor. If I am going to waste my time on this thread, how about a little background info?
For me: PhD 1981 in math, served on a PhD committee at Stanford on February 28, 2005, spent the summer of 2003 at a Max Planck Institute, spent three weeks at ANU earlier this year, have a paper which received a Featured Review, etc. (It does not mean anything but my PhD advisor's PhD advisor received the Abel prize last year.)

Also (La)TeX has become de rigeur for publishing in scientific journals. Compare the submission guidelines for sending an article to any The Physical Review journals in any of the TeX variants they prefer to that of doing a submission with MS Word. The American Mathematical Society apparently won't even accept papers typeset in anything other than LaTeX.

For scientific publishing, LaTeX really is the way to go.

The rubber band isn't a good analogy. The rubber half-sphere mentioned by g0dsp33d makes alot more sense to me.

I think the researchers used the rubber band analogy, it's probably because they are thinking about a Catastrophy Machine. I think this experiment might explain it, but it's been too long ...

Basically, pressure on the rubberband builds and builds and stops just at the verge of a big event. If something increases the pressure just a little bit, the rubberband snaps, and the circle rotates real quickly.

I am really impressed with the knowledge which /. moderators bring to the table. The parent of this post corresponds to the opinion of the majority of mathematicians, which is that Mandelbrot has done some good work but is a self-promoter, does not always acknowledge the work of others and probably is not considered a Math Legend by most mathematician. I am a peon with a Ph.D. in math who has been doing research for less than 30 years; I have only one paper which received a Featured Review and I consider myself to be a fairly ordinary person with three kids.
Slashdot moderators know much more than do I about contributions to geometric measure theory, Hausdorff measures, self-similar sets, fractals, etc. and I accept that my previous post was flamebait since I am ignorant. However, lots of other people deserve credit for their work on fractals. I will just mention two, John Hutchinson and Michael Barnsley. I just met John when I visited ANU recently; I believe I also saw Michael. I do not think they invented fractals or claim to be the smartest people in the world but they did some interesting work. Since our slashdot moderators are so smart, they can even tell you about this research. (Just in case they are too busy, here are some recent papers.)

And a bit more rigorously worked out here, for two slightly different waves interacting. Imagine more than two... http://www.ams.org/new-in-math/roguewaves.pdf/

Actually, Fermat's last theorem was proved about 10 years ago. It's highly unlikely that Fermat actually had a valid proof though.

The review by N. Geffen of "Analysis of Transonic Airfoils", Comm. Pure Appl. Math. 24 (1971), 841--851 by Garabedian, P. R.; Korn, D. G.

"Calculation of inviscid, subsonic-supercritical flow around prescribed airfoils is described. This supplements the authors' previous design of a shockless transonic wing using real and complex characteristics in the hodograph plane. The flow about the designed wing is calculated for a range of off-design conditions.

"Neumann's problem for the flow-potential equation is solved numerically in a plane where the exterior of the airfoil is conformally (also numerically) mapped onto the interior of the unit circle. Following E. M. Murman and J. D. Cole [AIAA J. 9 (1971), 114--121], a second-order finite-difference scheme is used in the subsonic region, while an implicit second-order scheme is used in the hyperbolic zone, introducing artificial viscosity of the right sign. The Kutta condition is satisfied by an iterative scheme. Results with relatively narrow shocks (i.e., steep gradients) are given and compared with wind-tunnel experiments."

If you look at Goldstine's publication list on MathSciNet (usually only available on University netwoks which subscribe), you find a strong collaboration with von Neumann and early uses of computers to solve scientific problems. For example:

"Preliminary Discussion of the Logical Design of an Electronic Computing Instrument" by Arthur W. Burks, Herman H. Goldstine, and John von Neumann, 2d ed. Institute for Advanced Study, Princeton, N. J., 1947.

"Planning and Coding of Problems for an Electronic Computing Instrument" by Herman H. Goldstine and John von Neumann, Institute for Advanced Study, Princeton, N. J., 1947.

"Numerical inverting of matrices of high order" by Herman H. Goldstine and John von Neumann, Bull. Amer. Math. Soc. 53, (1947). 1021--1099.

"Calculation of plane cavity flows past curved obstacles" by G. Birkhoff, H. H. Goldstine and E. H. Zarantonello, Univ. e Politec. Torino. Rend. Sem. Mat. 13 (1954), 205--224 (with a review by David Gilbarg).

"Blast wave calculation" by Herman H. Goldstine and John von Neumann, Comm. Pure Appl. Math. 8 (1955), 327--353.

(I mention Gilbarg's review in part because I had an office (approximately) across from his at Stanford for a semester in 1991.) Gilbarg's review begins
"This paper describes the first systematic calculations ever carried out on cavity and jet flows past curved obstacles, comprising altogether fifty symmetric plane flows past convex and concave bodies, under several different conditions of streamline detachment. The computations were performed on a high speed digital computer, using an improved version of a scheme previously described by Birkhoff, Young, and Zarantonello ... "

... shows that he's been offering "proofs" since July 1989. I see from MathSciNet that he has 87 papers from 1958 to 1994, but isn't this a bit like the boy who cried wolf?

If it turns out to be true, this will be super-duper-extraordinary - the man is probably in his 70s. G. H. Hardy wrote: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game". Wiles proved FLT at 40, Perelman of the purported Poincare proof is in his 30s... this is similar-level stuff. The only thing I can think of that even comes close is Fred Galvin in his 50s (?) proving the Dinitz conjecture.

You can follow discussions on sci.math and fr.sci.maths. Or read about how similar asymptotic proofs about properties of primes failed. Remember, this is arxiv - in the age of electronic preprints, you get many good proofs like Perelman's along with almost-proofs like Castro-Mahecha's and Dunwoody's.

If it turns out to be true, this will be super-duper-extraordinary - the man is probably in his 70s. G. H. Hardy wrote: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game". Wiles proved FLT at 40, Perelman of the purported Poincare proof is in his 30s... this is similar-level stuff. The only thing I can think of that even comes close is Fred Galvin in his 50s (?) proving the Dinitz conjecture.

You can follow discussions on sci.math and fr.sci.maths. Or read about how similar asymptotic proofs about properties of primes failed. Remember, this is arxiv - in the age of electronic preprints, you get many good proofs like Perelman's along with almost-proofs like Castro-Mahecha's and Dunwoody's.

First: considering that this site is supposed to be "News for Nerds" what news did the article provide? At a minimum it generated a forum to request further, more detailed information.

Second: Not unlike countless others, you misread one of my posts. I honestly can't see where I asked for theory. I am looking for how hard it was to solve this problem.

Wasn't that the point of the challenge? To quote the website: "to encourage research into computational number theory and the practical difficulty of factoring large integers."

So, with one of the problems solved, how difficult was it? 100 workstations and 3 months doesn't tell us much. Were all 100 working 100% on their given tasks for the full 3 months? Is it really 3 months, or was that number generously rounded up? How fast are the 100 machines? How do those 100 compare to the same number used to solve a previous value a few years ago? That last one is probably the most important question, because you can't compare between challenges without some honest reference.
And 100 is just an approximation - this announcement is just too vague!

As an example, something similar to this would have satisfied me. Please note that for that project, the delay between the date of submission and their detailed announcement is 1.5 months.

Please put a leash on your hubris.
I do. Perhaps you read frustration, and the expression of it, as arrogance. I've had difficulty finding such details, which is why I asked for assistance on the matter.

Although not quite what I was requesting, thanks for the links. It appears that Google has already provided a couple of them to me in the past. I've found this article to be an excellent primer - provides some history on (and some basics in) the effort of integer factorization. Most importantly it's not nearly as intimidating as the vast majority of publications I've encountered on the subject.

It's syntactically correct but meaningless.

I talked with NSA recruiters at the AMS meeting in Phoenix in January.

There are efforts underway to collect and streamline the (alleged?) proof of the classification of finite simple groups. See, for example, http://www.ams.org/online_bks/surv40-1/. I am not a group theorist (IANAGT :-) ), only a mere algebraic geometer, so I'm definitely not current on this issue, but I think everyone, and especially group theorists, are painfully aware of this huge steaming mess of a proof. They are working to bring it into a better shape.

There won't be any announcement; there'll just be books like the above-linked one that quietly turn up in mathematical catalogs.

It's not clear that any similar effort could even possibly be undertaken with a computer proof. You type in "RUN" and it spits out a long-ass proof. What are you supposed to do with that? Write a "elegant( )" function that makes proofs elegant?

This does not mean that computer proofs are bad. I guess my point is that a good, human proof brings greater understanding, which can lead to improved proofs, or better results, etc. Computer proofs seem to lack this feature of greater understanding. You get a result, and that's about it.

At least, that's the impression I've received over the years...

zach

Calculus volumes 1 and 2 by Tom Apostol (1967, 1969)
Mathematical Analysis by Tom Apostol (1974)
Introduction to Probability Theory And Its Applications volumes 1 and 2 by Feller (1968, 1971)
Principles of Mathematical Analysis by Walter Rudin (1976)
Real and Complex Analysis by Walter Rudin (1986)
Functional Analysis by Walter Rudin (1991)

My gosh...they are expensive!
But no way would my mathematics library be complete without them and others of similar quality.
They're OLD and EXPENSIVE, but they're the BEST.

Of course, for reasonably priced math books, you can't beat Dover publications.
The American Mathematical Society has some reasonable and excellent books in their bookstore., not to mention a few free ones as well.

Calculus volumes 1 and 2 by Tom Apostol (1967, 1969)
Mathematical Analysis by Tom Apostol (1974)
Introduction to Probability Theory And Its Applications volumes 1 and 2 by Feller (1968, 1971)
Principles of Mathematical Analysis by Walter Rudin (1976)
Real and Complex Analysis by Walter Rudin (1986)
Functional Analysis by Walter Rudin (1991)

My gosh...they are expensive!
But no way would my mathematics library be complete without them and others of similar quality.
They're OLD and EXPENSIVE, but they're the BEST.

Of course, for reasonably priced math books, you can't beat Dover publications.
The American Mathematical Society has some reasonable and excellent books in their bookstore., not to mention a few free ones as well.