Books to Buy for Mathematical Maturity in Theory and Proofs

[blunt object]

I hope this is in the right section, otherwise please move it. This might be a long post and I hope someone will give it a shot. It is about books but not really a comparison between them. I will give some background first, so you might skip to the end (- Books * -) if you are only interested in the title question. With graduate I will mean masters level and above although I am generally used to bachelor and master being undergraduate and graduate meaning PhD students.

- Background -

I am a masters student in engineering. Focus is generally on computation and deriving formulas (theory) but no really what mathematicians call proofs which is something I am not familiar with the slightest. Some of what we do and use in our courses includes: tensors, FEM, CFD, control theory to make the list very short. So I do know some (some might call it a lot) of applied mathematics.

I would like to be able to study Functional Analysis, higher level PDE (well the analysis part of it) and some mathematical aspects on classical mechanics to name a few subjects. These are generally courses given by the mathematical institution belonging to their masters program. As such they are mathematically rigorous, at least more then I'm used to. A typical textbook can be theorem, proof, trivial example, theorem, proof and so on. The exercises are mostly show and prove this and that. The prequisitions listed for these courses are Calculus and Linear Algebra which is something I'm quite accustomed to (the computational part of them at least) and passed with good grades. But I believe it's a little misleading.

- Books Studied -

The book used in our Calculus courses was:
Calculus: A Complete Course, Robert A. Adams
Whole book used with the Linear Algebra book and other lecture notes over three courses. Proofs of trigonometric stuff, induction and that sort was covered in those courses but it was three years ago and it was not in the "theorem, proof" styling.

The book in Linear Algebra courses:
Linear Algebra and its Applications, David C. Lay
Whole book used over two courses with the Calculus book.

Lecturers notes on Transforms (Fourier, Laplace, with applications to ODE, PDE).

Lecture notes on tensors.

ODE, PDE have been introduced a lot trough mechanic courses and sometimes had some rigour (solutions belong to this and that space with set notations, not really proofs and really brief).

Statistics (and probability) course (starting with basics and ending with basic linear regressions).

I feel that I'm not really prepared to take on, say, Functional Analysis as the professor is basically rambling up axioms, theorems and proofs on the black board. The book used is: Introduction to Hilbert Spaces with Applications, L. Debnath, P. Mikusinski.

- Books to Buy -

What I want to learn is mathematical rigour, logic, notation (set notation mostly, I don't know if a book on Set Theory is really what I want though) and just gain mathematical maturity. I guess a good starting point would be to have exercises on material I know but that asks to prove this and that. Although I would like to learn some new area along the rigour way to do mathematics. Right now I just don't see that a proof really proves anything and I'm even less likely to prove something my self. This has lead me to the following books. I want a book from both areas. I cannot stress this enough, much logical reasoning and all the standard notations for proofs are something I want in the book and of course figures to illustrate the theory are welcome.

Calculus books by Spivak or Apostol (both volumes). Maybe Courant, but what or how many volumes? I'm leaning towards Apostol as it covers multivariable calculus although I don't care much for the Linear Algebra part (see why below). I have read most posts on PF about these two books and they both seam great. What I can't decide on is if there is any point in buying these. Might it be better to buy something like: Analysis, Steven R. Lay. I can't say I'm really into "Real Analysis" although I might want to take it on some day but for now I want to gain maturity and learn all about proofs and the used notations (sets, subsets, and all those notations you could think of). From what I can see (brief look in the library) neither Apostol nor Spivak use for example set notations in their proofs but that might belong more to Linear Algebra.

Linear Algebra books by Friedberg or Hoffman and Kunze. I'm leaning towards Friedberg as I believe it's enough (The Hoffman and Kunze might be to pure for what I'm seeking). I don't know if it will cover any new material (not needed) but from what I have heard and if it is anything like Hoffman and Kunze (which I have glanced at) it is rigorous.

- End Notes -

So, was my Calculus And Linear Algebra books pure **** and should I refresh my knowledge with more rigorous books or move to some introduction to real analysis? Do you guys have any ideas on other areas to study, books to self-study, before taking on a Functional Analysis course with the mentioned book above or would it be enough to gain some maturity from some (in that case which) of the mentioned books on the buy list.

I was thinking of reading trough: Book of Proof, at least the Fundamentals which should be enough on logic and sets.

Well budget is not infinite so one (all volumes) in each area mentioned (or the Real Analysis mentioned).

I want books that have at least solutions for half of the exercises. Not books for which answers are impossible to find.

- No Need to Read -

I don't want, nor do I have the intention, to do Calculus my whole life but as an engineer I feel that this is so much more useful then say number theory. And for PhD Studies later on it might be good to have maturity as I guess control theory articles can use their fair share of math. More Probability and Statistics is something I plan on taking later when I have developed some maturity.

 

[micromass]

I know you are eager to start learning something new, but I fear that might have to wait for a while. I think it is best to start doing rigorous versions of things you already know. This will already be challenging enough.

I suggest that you start by reading that "book of proofs". It looks quite good. Other proofs books are available, but they aren't free.
Other books might include:
https://www.amazon.com/dp/0486453065/?tag=pfamazon01-20

which is quite good and cheap or

https://www.amazon.com/dp/052171978X/?tag=pfamazon01-20

which is my personal favorite. The most popular proof book is without a doubt: https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

But anyway, if you like your free book and you can understand it reasonably well, then study it!

The next step is linear algebra (because I find it easier to learn how to do proofs in linear algebra, as opposed to proofs in analysis which are quite complicated). The Friedberg book is very good. Another very good book is Lang's "introduction to linear algebra" as well as his harder "Linear algebra". I don't recommend Hoffman and Kunze, as it's probably too hard and too abstract for now (it IS a very good book though).
Since your interest is in functional analysis, I also want to recommend Halmos https://www.amazon.com/dp/0387900934/?tag=pfamazon01-20
The focus here is on (finite-dimensional) linear algebra, but he actually adapted the proofs from the infinite dimensional version. So the techniques you learn in this book, might be applicable later. (as opposed to proofs which use determinants, for example, which do not generalize to the infinite-dimensional setting).

Then there's analysis. You probably never did rigorous calculus/analysis proofs like epsilon-delta things, so this is definitely something you need to master. Like you said, Apostol and Spivak are very good books. They are called calculus books, but in reality, they are quite rigorous (introductions to) real analysis.
Somehow, I'm a big fan of Spivak. And I'm certainly not a fan of Apostol's multivariable book, I think it's a bit weird. A very, very good book on multivariable "calculus" is https://www.amazon.com/dp/0130414085/?tag=pfamazon01-20

Then you can start with functional analysis. I think that the Debnath-Mikusinski book is really great, but it might be hard for newcomers. Especially his chapter on Lebesgue integration is a bit confusing.
A very good book on functional analysis is https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20
This book was my first book on the subject and I found it really nice. A negative point is that it doesn't cover Lebesgue integration, but that might be a good think because it's not easy for people new to the subject. In contrast with other functional analysis books, it also does not require topology.

If you're interested in learning measure theory and Lebesgue integration, then https://www.amazon.com/dp/0763717088/?tag=pfamazon01-20 is a great and intuitive introduction.

Some of the books I recommended here might be too hard for you. Do not buy the book before reading a few chapters first. If you think that after reading the first chapters that the book is too hard for you, then you can always ask for easier books. (If you want, you can always PM me).

 

[Jorriss]

If you are interested in classical mechanics, take a look at Arnold, Mathematical Methods of Classical Mechanics.

 

[blunt object]

Thank you for a thorough answer. It is no coincidence that I suggest books that you find good, as I have searched around quite a bit and read many posts from PF and at last found your blog posts on the subject. Not that your opinions are all that mater of course :), but no one seams to be talking bad about these books in general and I guess I won't go wrong with either one.

Thanks for the suggestion on "proof" books. I'll pick two as they aren't to thick and quite cheap. I don't want to be stuck on proof books for to long to no end either.

Interesting choice, on choosing Linear Algebra before Analysis, I guess it makes sense. I have not been able to read anything from Friedbergs book (can't find any "peaks" into it from amazon and sites as such). But somehow I get a feeling it gives a more complete overlook in one book as it's thicker then the other (I could very well be wrong here). That it's more suited for a course (or self study of the material, self-contained and gives more explanation). Somehow I have also created the impression that it ought to be easier then Lang and Halmos although you make an interesting point about how Halmos adopts his proofs. I don't know about Lang as I have heard some bad comments about him overall (his style). Would you recommend any of these three books over the other and how would you order them in terms of difficulty?

<Edit>
Halmos book doesn't seem to have any set notations as I just briefly went trough it so it's out of the questions. As most of the books used in our math masters program use sets notations very much, too much, (as do the professors) I would like to get used to it right away.

Anybody know how much set notation Friedbergs book uses? Linear Algebra Done Right by S. Axler seems to incude it a lot which is that I want. But I have also read a lot of criticism about it (a lot of good stuff to though). I could not go trough Linear Algebra by S. Lang to get a clear picture of how much set notation he includes. Lang's "first book" on the subject (Introduction to Linear Algebra) does not seem to include set notations at all and I'm afraid it's the same with the book mentioned above.
</Edit>

I have already been reading a few pages from Apostol and Spivak and I could not decide on either, they both seemed good. I let the factor be that Apostol cover multivariable in the same "series" but this is the second time I hear someone not liking it (vol. 2) that much and as I don't care for the LA part in it I might as well go with Spivak as it's cheaper. I guess I can hold off on multivariable as I know its applications and how to use it. Might it be enough to do "proofs" in one variable and move one when (if) I understand it thoroughly? But I plan on getting to it as well.

I don't have much of a choice on the Functional Analysis book. I have been looking at that particular Functional Analysis book you mention and might get it (later) just to read a little on my own as it seams to be more gentle then the one used as course literature at my University. I have read a little from it on metric spaces and it's a much better introduction then the book I got in my opinion (from the little I have read from both books). I find it somewhat strange that we can take the course and they know (or maybe they don't) how badly prepared we (from engineering) are.

<Trivia>
I am quite interested (to learn) in measure theory and Lebesgue integration and I believe you that the book is good but that is so far into the future that I'll have to look it up later. I'll come and find you here when (if) the time comes :).

As the course in Functional Analysis is given one year from now I hope it will be enough time to learn. I'll read some more from the books before deciding. I don't think an easier book (in say calculus or linear algebra) would benefit me as I do know the material quite well and I have to start with proofs somewhere. I have been looking at my University's bachelor program in math and they have the same Linear Algebra book I had but then they also have a second course with lecture notes (I guess I would have to attend lectures to get the full of these short notes). They also have an introductory course on logic, set notations and catching up from high school (well it's not called that here but so everyone understands) and I guess this is what I'm missing along with a better book in calculus.
</Trivia>

If you are interested in classical mechanics, take a look at Arnold, Mathematical Methods of Classical Mechanics.

That book is what I had in mind but I have a feeling it requires more rigour from me. I have been dealing with Hamiltonian and Lagrangian equations to some extent in structural dynamics (applied) but I don't know how much "proofs" he emphasizes, as it is possible to take a PhD course with that book at my University. I guess the course might not bee "PhD level" but usually they put upper level courses (graduate) that few students would otherwise attend as PhD courses and they are given when enough students want to take it.

 

[micromass]

I forgot to include Axler in my previous post. Since you know linear algebra already (the computational variant that is), I think Axler would be a very appropriate book to study from. You already looked at it and liked it, which is very good.
The problem I have with Axler is that it doesn't do determinants, which makes in unsuitable for a first course. However, you already know linear algebra and functional analysis doesn't make use of determinants anyway.

If you want to learn set theory notation, then I would certainly go for Velleman's proof book. It includes a lot of set theory. I learned to be very proficient with set theory from topology and measure theory courses. These books are filled with set theory. Unfortunately, you don't need to go through topology. Maybe going through measure theory could help.

 

[mathwonk]

halmos has a nice little book called "naive set theory" which many people of my generation learned from. after that you essentially never need to study that subject again.

http://www.abebooks.com/servlet/SearchResults?an=halmos&sts=t&tn=naive+set+theory