Comment by bumblingmumbler | Hacker News Reader

Comment by bumblingmumbler | Hacker News Reader

bumblingmumbler

I am almost always disappointed in such lists as they almost never contain the books I have never seen before. I got lucky today.

The list has [0] Multivariable Calculus by Don Shimamoto. I saw it on Amazon first(so I have seen it before, but not on lists such as the linked one), but didn't want to pay $40 for this strikingly beautiful book. Turns out it's free.

The real analysis section has [1] by Lafferriere, Lafferriere and Mau Nam. It's short, sweet and very pretty. No bloat. Logically organized. For example, the section "Applications Of The Completeness Axiom" contains everything you need to know as pertains to the topic (within elementary analysis) in one place. Such things usually get lost in the weeds in regular analysis books. Another book I like there is [2] which is on Lebesgue Integral and Measure Theory by Cheng.

[0] https://drive.google.com/file/d/1IB-fSe-KR6mbo89iFTJVSA0CrFf...

[1] https://pdxscholar.library.pdx.edu/cgi/viewcontent.cgi?artic...

[2] https://www.gold-saucer.org/math/lebesgue/lebesgue.pdf

> The real analysis section has [1] by Lafferriere, Lafferriere and Mau Nam. It's short, sweet and very pretty. No bloat. Logically organized. For example, the section "Applications Of The Completeness Axiom" contains everything you need to know as pertains to the topic

I guess this is a matter of taste. I went to look at this and it really bothers me that, true to the name they give, the "completeness axiom" is presented as an axiom. My analysis class used Strichartz' The Way of Analysis, which is certainly not "short and sweet" -- the professor described it to us as "chatty" -- but it goes ahead and actually proves the "completeness axiom".

Compare the completeness axiom

> Every nonempty subset A of ℝ that is bounded above has a least upper bound. That is, sup A exists and is a real number.

with Strichartz' theorem 3.1.1

> For every non-empty set E of real numbers that is bounded above, there exists a unique real number sup E such that

> 1. sup E is an upper bound for E

> 2. if y is any upper bound for E, then y ≥ sup E

Maybe Laferriere, Lafferiere and Mau Nam could use a little more "bloat". :/

In the first one, they already defined upper bound and least upper bound beforehand, so the statement is more concise than the Strichartz's one. On the other hand, the proof of the completeness axiom, or rather a detailed construction of the reals, is a part I always enjoy in analysis textbooks.

Yes, I have no problem with the fact that the concept "sup A" is defined before the statement of the axiom and then not repeated in it. (It's defined before the statement of theorem 3.1.1 in Strichartz too, it's just also repeated there.) I'm objecting to the fact that they present a theorem as if it were an axiom.

One model's axiom is another model's theorem.

Sure, the proof in Strichartz is constructive, defining a real number that is the supremum of the set.

LLN (I'm going to assume that the family name is Nguyen) can't do that, because they have no model of the real numbers, so it isn't possible to say that some construct is or isn't a real number.

So this is more a case of "what is a theorem if you have a model is only an axiom if you don't".

If you changed the statement of the completeness axiom from "every nonempty subset A of the real numbers that is bounded above has a unique real least upper bound sup A" to "every nonempty subset A of the rational numbers that is bounded above has a unique real least upper bound r", you'd have the Dedekind cut construction of the real numbers. That's not generally presented as an axiom either.

There are sometimes good reasons to define a mathematical object under study (vector space, real numbers, ...) by an axiomatic list of properties it satisfies instead of constructing it from simpler objects. John Conway's ONAG has an interesting mini-chapter about this.

How does Strichartz prove it? You certainly need some axiom beyond, say, the ordered-field axioms; for example, the analogue of Theorem 3.1.1 isn't true for ℚ in place of ℝ.

Strichartz constructs two Cauchy sequences (an increasing sequence starting with a point in E, and a decreasing sequence starting with a point that bounds E above) defined by successive midpoints between the points in each sequence. (That is: take the midpoint between the current top of the increasing sequence and the current bottom of the decreasing sequence, and assign it as the next element of the increasing sequence if it is not an upper bound for E and otherwise the next element of the decreasing sequence.) You can show that

1. These are in fact both Cauchy sequences;

2. They have the same limit;

3. That limit is the supremum of E.

LLN has no definition of a real number, and therefore can only construct real numbers by field arithmetic on other real numbers. That's what's missing.

(I'll take a moment to tangentially observe that I like how my description of the proof neatly explains why it requires the assumptions it does: the theorem requires that E be non-empty because the increasing sequence starts from a point in E, and it requires that E be bounded above because the decreasing sequence starts from an upper bound for E.

If you demand that the sequences start from rational numbers, it's easy to fix the upper bound, but you have to relax the start point of the increasing sequence from "a point in E" to "a point that is not an upper bound for E". That still requires that E is non-empty, but it does it in an awkward, roundabout way.)

Yes, fair enough: it fits more naturally into the general context of metric spaces to come to postulate the completeness of ℝ as a metric space and prove its completeness as an ordered space, rather than the other way around. Of course, one still does have to have a completeness postulate in some sense!

Theorem 2.3.1 (Completeness of the Reals) A sequence x_1, x_2, ... of real numbers has a limit if and only if it is a Cauchy sequence.

(The wording is unfortunate in this context as stated, but the proof shows that the limit of a Cauchy sequence of reals exists and is itself a real number as opposed to some other kind of thing.)

Completeness is something we want; a lot of effort goes into proving that we've actually achieved it.

The fact that Theorem 2.3.1 is given a name does suggest, though, that Strichartz agreed with you as to which kind of completeness was more significant.

I was about to ask how he proves that, but that way lies an infinite regress, so I found a copy and checked. I see that he defines (without using the terminology) the real numbers to be the completion of the rational numbers for the usual Archimedean metric and so, as you say, is able to prove the completeness rather than postulating it. (This explains to me your remark elsethread (https://news.ycombinator.com/item?id=22712940) that "So this is … a case of 'what is a theorem if you have a model is only an axiom if you don't'.", which I didn't understand on first reading.)

Note that this is just a difference of terminology from other approaches; those approaches prove properties of an abstract object satisfying certain axioms, and then are obliged to exhibit a specific such object. The approach you favour is definitely more hands-on, but runs the risk of obscuring that really field axioms + uniqueness is all we need; if someone else prefers to construct their reals using, say, Dedekind cuts, then there is no need to check all the theorems proven along the way, only that the axioms of a complete (as an ordered or metric space) ordered field still hold.

Disclaimer: please, excuse the typos, grammatical and stylistical mistakes below as I am pretty freaking drunk at the moment.

Friendly reminder:

while the material in the linked books might appear trivial to many, the style might be discouraging to those who've never seen "actual math" before. If you're in the latter group, try out the books below (some are not free, but they're still helpful [libge*n might be helpful]):

1. BOOK OF PROOF by Richard Hammack. This FREE book is your passport into mathematics proper. It teaches you how to read math books. After this book you'll stop fearing the word "proof" and start demanding it wherever you go.

Link: https://www.people.vcu.edu/~rhammack/BookOfProof/

2. The book by Don Shimamoto is super trivial if you know linear algebra. The best books in the genre (imo) are (a) Linear Algebra: Step by Step by Kuldeep Singh whose secret is that he uses all the exercises from other LA books as his 100% worked out examples (which seems to be a tradition for Indian authors; I have tons of examples from other branches of math). Indian authors teach you how to drive a car by putting you at the steering wheel instead of lecturing you about it. BTW, if you know math at the level of, say, 7th grader, then you're perfectly suited to study from Singh's book. If that isn't clear, do you know how to add and multiply fractions? If yes, then you're perfectly ready.(b) A modern Intro to Linal by Henry Ricardo. This book is great in its ability to explain certain introductory concepts. For example, the linked Shimamoto book starts out with (going by memory) "the elements of R^n are finite sequences of length n... bla bla bla". Well, Ricardo explains what that means. It also helps if you know what a "sequence" is. But we're getting there. Another author who does a good job of it is Sheldon Axler, I believe. He does have his own idiosyncrasies, but I digress. The only difficulty here is, probably, the idea of diferentiability of multivariable functions.

Links:

https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh-ebo...

https://www.amazon.com/Modern-Introduction-Linear-Algebra/dp...

http://sites.msudenver.edu/wp-content/uploads/sites/385/2017...

3. Now to analysis of real kind as opposed to the complex one. If you've never heard anything about this, then you should hook up with Lara Alcock's "How To Think About Analysis". To the best of my recollection, half the book is about how to think about the term "sequences". She goes above and beyond in trying to explain you what that means. The rest of the book is so-so. Once you learn about sequences, you can move on to Jay Cumming's book. It's a bit more involved with proofs and shyt, but still extremely accessible. Laughably so. But note, he only stays grounded on the real line and does not venture into the Euclidean Spaces proper. However, there's another easygoing dude named Raffi Grinberg who'll take you through R^n by hand, no problem. If you master this book, then you're at the level of what's referred to as "Baby Rudin". Next, if you're feeling adventurous, there's a new book by Sheldon Axler that introduces you into the modern analysis which is FREE FOR NOW. GRAB IT WHILE YOU CAN CAUSE I SAW IT ON AMAZON.

Now if you're a business type who ain't got no time for all this foolishness, try "Intro to Analysis: From Number To Integral" by Mikusinski's. It starts with axioms for reals and takes you straight through to Lebesgue Integral in no time. Every section has very few exercises. Say, about 4. It also skips through the traditionalist Riemann integral. Because why not.

Links:

https://www.amazon.com/Think-About-Analysis-Lara-Alcock-eboo...

https://www.amazon.com/Real-Analysis-Long-Form-Mathematics-T...

https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

https://link.springer.com/content/pdf/10.1007%2F978-3-030-33...

https://www.amazon.com/Introduction-Analysis-Number-Integral...

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Now to a few examples of Indian authors' propensity to give you everything you need to know about the subject unabashedly. Check out:

-- "Sequences and Infinite Series" by N.P. Bali. It's 100% worked out proofs after proofs after proofs. If you read through this book, I imagine you could easily crush any "sequences and series" portion of any elementary real analysis exam in existence. The author also has a book on real analysis which is structured very similarly to the aforementioned book. No idea what N.P stands for, though.

Now to a book you've never heard of. "Data Structures and Algos" by Gav Pai. It easily blows Cormens, Sedgewicks and Skienas out the park by giving you, as you guessed it, all the 100% worked out examples you need to understand the topic.

"Fundamentals of Complex Analysis by Bhat" is another one. All its problems are immediately followed by solutions. Look, it understands you're not at 40K a year school with 10 profs to help you out with any of your questions. Books like this are written for students, not to impress the profs.

I just remembered some other Indian textbooks on abstract algebra, discrete math etc, but I cannot remember their names. They are also all about giving you all the examples you need to grok the concept at hand.

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