Don't expect anything better here.

First, let me go off on a diatribe of what I like in mathematical writing.

I...am of a rare breed. I will say something that would get you lambasted in nearly any polite snobbish discussion of mathematical literary criticism. The kind that takes place in closed backdoor rooms with tea, coffee, and cookies.

I

Like

SHOCKING Bourbaki SHOCKING

I love the style of the Bourbaki books, I even think they are fantastic to learn from. Really, in my mind, mathematical literature takes two stances. Two superpower opponents standing on either side of a hill. One championing butter side up, the other championing butter side down; one is red, the other is blue; one white, the other black; good vs evil; Pepsi v. Cola; vi vs. emacs. This distinction is to me: discussion style vs. Bourbaki style. The discussion style is relatively straightforward to explain and nearly 90% of mathematical texts use it in varying degrees. The discussion style is exactly what the name implies: the mathematics therein are expounded upon in a linear, verbose, talkative manner. The theorems and definitions are provided as necessary, with examples, counterexamples, historical treatments, expository remarks, etc. taking place in a haphazard fashion immediately afterwords. There are usually no labels for definitions nor theorem nor remarks, they occur in a jumpy matter. It is as if, and this is no surprise, you are reading what you usually encounter in a lecture. This is somewhat expected, as the vast majority of mathematical texts are created or adapted from lecture notes of various professors. And, like I said, this is taken to different degrees. Some mathematical texts approach the subject from almost a completely historical angle. Noting the various characters along the way and using it to guide development. Some are incredibly rigouris and teach a train of courses that interlock in a long interwoven web. Schechter and Deuidonne champion this sort of literary style. Deuidonne especially so.

Before explaining the Bourbaki style and getting into why I like it, let me finish my promised diatribe against this discourse-style trite. The point of a mathematical text is, to me, one of two cases: to use to learn from, or to use to reference from. The second is an ample point that brings lots of easy targets to bear against discourse-style literature. As we often do in mathematics, let's look at the extreme cases to analyze the faults and problems at hand. Imagine a book which is entirely discussion centered. There are no symbols, no formulas. Every formula is instead explained in a longwinded talkative manner. There are no chapter outlines either. The entire book is a solitary, gigantic, long novel. Now that I have created this setting, I will answer the original question via the rhetorical question: do you really think this would make a good reference? Even if there was a complete and methodical index, you would switch to the page (or, in this extreme case, the set of dozen of pages), and you would still have to do a lot of nontrivial reading to decipher where is the item you are searching for as a reference piece. Furthermore, any piece of analysis is likely to continue for tens or hundreds of pages longer than necessary. Which brings me to the second point: would this be useful as a book to learn from? I contend that it would not. Can you imagine the time it would take to read through what, in the extreme case discussion-style mathematical book, might be hundreds of pages for what is normally a ten line theorem? The unnecessary trite of historical discussion and examples? Yes, it would be complete. Yes, perhaps after spending a few days reading such trite you would understand Arzela-Ascoli much more than anyone else. BUT, the time involved would be much more than if you had actually worked with the theorem at hand. And the relevant information would be, and this is probably the most important point, spoonfed to you. You would garner no useful concept of the function of the theorems and concepts at hand. Yes, this is what exercises are for. But keep in mind that both the Bourbaki style and discussion style book have them. And I contend that the Boubaki style is much more apt at producing a student who can do exercises, and gain a self-made intuition of the object at hand. Due to not having to go through the unnecessary expansion of the material at hand read, by simply having more time for himsefl to do so, and by not having to go through unnecessary reference slogging. With these points already hinting at my prejudices for the Bourbaki books, I suppose I should move on to...

What is Bourbaki? And why do I like it so much? Bourbaki is essentially the minimal amount of material necessary to know the subject. It is a brutal definition-statement-theorem-statement-proof exposition. There are no remarks (or hardly any, really), there are no historical details, and there is no intermediary discussion. And, in my opinion, the best example of a Bourbaki book was not by the Bourbaki book, but Sir Atiyah's Commutative Algebra book. As I've already alluded to, it makes an excellent reference book, because the exposition is so concise, and so well organized an labeled that it is very easy to recall, organize, and remember material. As for learning from the book, I suppose I should also make a side remark on teaching styles. Since, as I alluded to before, a lot of books are written in accordance to lecture notes created by the professor of the subject at hand, the real questin boils down to what are the preceding methods of teaching that are respectively substituted for the literature styles? Well, the discussion centered teaching style is very simply explained as the traditional lecture-HW teaching style. And the Bourbaki style, I would claim, aligns perfectly with a Moore-method course. Why? Look at the description of how any Moore method course begins. They hand out a list of the definitions, theorem statements, and corollaries. You provide the proofs. Or, in later years, maybe a few proofs were provided, but for the most part the entire course was a gigantic exercise in exemplary mathematical thinking. And to connect the dots, what exactly is a Bourbaki style book again? A Bourbaki book is that concise sheet handed out on the first day of a Moore class. In other words, my claim and conclusion is that Bourbaki-style books line up perfectly well with Moore style teaching. And this, essentially, is why I also think Bourbaki books ar great to learn from. Because the examples, the remarks, the counterexamples are not spoonfed.

Now...I suppose the reactionary discussion-style supporters would note that the above diatribe was not written in a Bourbaki-like fashion. I would reply that this is because it actually takes time, diligence, and care to write in such a clear and magnificient prose such as Bourbaki. That is, if this website was written in such a manner, it would only improve its quality. The only reason I don't, is to exemplify the poor quality of the former method.

Either way, like in my "How to Latex" page I got carried away in a discourse on emacs vs. vi, I seem to have gotten carried away on a similar discourse here...my apologies. I feel like Sylvester at this point.....

Dieudonne seems to have the opinion that the whole of mathematics should be the mastermind of one particularly senile old geezer talking non-stop for a few decades. And he accomplishes just this in his books--I MEAN BOOK. The beginning is too trivial and trite for anyone to have the wherewithal to stand for, and by the second page he accomplishes the nearly impossible task of being impossible to read without having read the first. Unlike some authors who seem to have this silly notion that a student in analysis should be able to pick up a more advanced book in analysis having understood the simper concepts, Dieudonne does away with these notions and assumes that everyone must be at a clear slate. Examples are also a thing far too trivial for him. After all, everyone learns from the abstract to the concrete, MIRITE?

Alright, let's look up the proof of Alaglou. Wait. Why is the whole proof only three words long? All of Gelfand theory only takes a page? How is this book 200 pages long?

OH NO, THE AUTHOR IS STILL ALIVE AND IS AT PURDUE. I'M DEAD D:. BETTER SAY SOMETHING NICE Why do I like this book? By all my technical aspects and the rambling foam up top, it sucks. It's discussion-based, rambling, conversation style. And yet, I really like it. It has pretty good explanations. It approachs numerical analysis from a THEORETICAL POINT OF VIEW, unlike this Kincaid and Cheney piece of--

...Really? Really? Are we suddenly retarded undergraduates now? Doing ditto exercises and pretending we know group theory by asking us to show that Z_3 is cyclic (and giving a hint, at that?)? This is a retarded book meant for retarded undergraduates and average high school students.

See Gert Pedersen.

There is a reason this is referred to affectionately as "Baby Rudin". This is, honestly, what every undergraduate course in real analysis should be using. This is a real introduction into mathematics. And, it is hard to think of a way it could be made better. The latest edition has a discussion on wedge products that the author of these reviews finds particularly amusing.

There is a reason this is referred to affectionately as "Papa Rudin". A beautiful companion to its baby-ancestor. Honestly, if a mathematician were brought up on Baby Rudin when they were young, and next on Papa Rudin, I could hardly see any difficulties in analysis that would fall bereft of such an individual. The exposition of "Papa Rudin" as a whole now-a-days may be a bit odd, but the exposition within each chapter and the exercises within are superb.

Lang is a prevalent author. And his opinions on mathematics are vast and wide, and his scope seems to be ambitious: to write on all of mathematics. He has accomplished this, bringing in nonsense where it should not be displayed, and making the whole of mathematics unorganized and extremely difficult to learn from. His problem, perhaps, is that his audience, net, scope, and ambition is too large. Like the parable of the dog reaching for the meat seen in the river water's reflection. The whole becomes an unstandard mess with which to learn any kind of mathematics from.

There is a happy conversational style taken in the book. Perhaps helpful if you are bored, locked in your room, have nothing to do, already know functional analysis, and wish to hear what someone who has had the equivalent amount of boredom has to say about functional analysis. Other than that, the conversational style makes it impossible to learn from, impossible to reference from, and thus impossible to be anything close to useful of a book. The approach is standard, the work tacky, and altogether completely mediocre.

Seriously, how does the AMS make its books? It sounds like a member of the AMS stashes a recorder on his person, goes into a fairly talkative mathematician's office, asks him how he would teach the entirety of a subject; and then, when the AMS member leaves the office, he goes and plays stenographer with the recording he just made and then use some underhanded copyright rules to make the author agree with all of this.
What I am saying is that it is another one of this 'conversational style' quote-on-quote 'masterpieces'. Where the style, although interesting in seeing the author's opinion and views on the formulation of the subject, leave the book without any utility otherwise.

Was this guy on crack when he was writing books? Seriously, what in the world is this? The ONLY thing good is a small discussion on Littlewood's Principles in the third edition. If you are seriously working your way through anything before the fourth edition, read the third edition's section on Littlewood's principles (maybe...maybe, if you are curious, see how he thinks of functional analysis in the context of measure theory), and then promptly throw away the book.
Into a fire.

Having laughed off all other editions of Royden, I was surprised upon hearing the uproar about his fourth edition. And, I was so incredibly surprised. The exposition as a whole has been made beautiful. Within each chapter it rivals dear old "Papa Rudin". And...I'm afraid I must hold back on advice for the exercises until I have tried them more deeply myself. I was impressed. Very, very impressed. This may be the best book in graduate analysis that is available as of now.

A God-send. A breath of fresh air. Here was a man who was a communicator of mathematics. Every book is written well, both to learn from (except maybe the first edition of some of his earlier books, if only because it is lacking the exercises; but the Hilbert Space Problem book corrects for this defect), and as a reference. The style is not tacky, not too conversational, yet certainly pleasant to learn from. Everything is labeled should you need, everything is exposed in a thought-provoking way. Especially interesting, oddly enough, are his writings that are not from mathematics: I Want to Be a Mathematician, and his articles about how to read, write, and speak mathematics. These are classics and, in my opinion should be read from everyone in the field. I might add another caveat for Naive Set Theory, here the style is overly vague and conversational; however, this seems to be a problem from the philosophy of mathematics, rather than the author himself.