Learning from stewart and then from spivak

>not an argument

>I have 6 months to decide what to study and wanted to try some book mathfags study in college
>I have no experience with proofs

Read a book on proofs and/or one of these:
Veeky Forums-science.wikia.com/wiki/Mathematics#Overview_of_Mathematics

This is the only non-meme list for a beginner mathematician:

>Chapter 1: Introduction to Entry-Level Mathematics, P. I
Pre-Calculus - Carl Stitz & Jeff Zeager
Calculus: A Modern Approach - Jeff Knisley & Kevin Shirley
The Art and Craft of Problem Solving - Paul Zeitz

>Chapter 2: Introduction to Entry-Level Mathematics, P. II
Linear Algebra and Its Applications - David C. Lay
Calculus of Several Variables - Serge Lang
Applied Differential Equations by Vladimir A. Dobrushkin

>Chapter 3: Introduction to Proofs and Survey of Higher-Level Mathematics
How to Think Like a Mathematician - Kevin Houston
How to Prove It - D. J. Velleman
Mathematics: Its Content, Methods and Meaning - A.D. Aleksandrov, A.N. Kolmogorov, & M.A. Lavrent'ev

>Chapter 4: Bringing It All Together: The First Test of Mathematical Maturity
Calculus Vol. I & II - T. M. Apostol
Analysis I & II - Terrance Tao

Nobody gives a shit about your personal reading list. No one list fits all.

Good books in this list

>>Go to a library, in best case a university library.
>>To the calculus/analysis shelves.
>>Take all different books out and look up their table of contents and read some pages.
>>Keep whatever book you found most appealing to you.
If it is Spivak, then Spivak is fine.
Don't take a look at one book exclusivly because you read it somewhere, always check out what style fits YOU best.

>thinking that people aren't capable of picking and choosing what they need
>not understanding that the most important part of a list is flexibility

>Linear Algebra and Its Applications - David C. Lay
>not Stephen Andrilli, David Hecker-Elementary Linear Algebra-Academic Press

close but no cigar

Browsing TOC, previews, and MAA reviews makes me like Hecker, but I'm not convinced. Give me 3 reasons to use Hecker over Lay for a first course introduction.

Spivak and Stewart teach different things.

Stewart teaches a bunch of general computational techniques so that you can use calculus in most general contexts. Stewart's book is disliked by pure math students because it really demands that readers have an intuition for lots of obscure tricks and specialized techniques (many of which can not be applied in the sort of general setting one tries to deal with in pure mathematics).

Spivak on the other hand teaches a babby version of analysis which is essentially an introduction for how one can formally reason about calculus from a more theoretical perspective. Spivak is not necessary for most applied math people as it builds up a lot of theory and formalism that doesn't really matter to applied math and other areas.

Personally I don't like Spivak and believe there are better books and lecture notes out there. In my opinion, Spivak is too informal to really be a good introductory analysis text (probably why it's named Calculus instead of analysis).

What is a better version of Spivak?

Clark's book was supposed to be inspired by Spivak but with improvements: math.uga.edu/~pete/2400full.pdf

Another good text is by Knisley, though it's more like Stewart and less like Spivak in scope (but better in intuition): faculty.etsu.edu/knisleyj/calculus/final.pdf

another meme list

Everything is a meme

Just dive in nicca

Why are these lists so fucking long and full of crap. Just read Rudin, HoffmanKunzeAxler, HersteinDummitFooteArtin, Spivak's calc on manifolds. If it's too hard then read the corresponding section from something easier or just give up. After you've done all that read this usamo.files.wordpress.com/2017/02/napkin-2017-02-15.pdf

>usamo.files.wordpress.com/2017/02/napkin-2017-02-15.pdf
this looks ambitious to say the least, hard to imagine who the entirety of this thing would be useful for but i guess taking it in bite-sized pieces could be beneficial

Learn from Simmons then from Spivak.

Or directly form Spivak.

DURR JUST START WITH REAL ANALYSIS

autists

Everybody except for the most autistic reddit-tier mathematicians seems to love this list. What gives?

>everybody loves it
>list has only one positive reply
keep reposting your meme list but it doesn't make it good lol

It's not relevant since you're a burger, but there's this semi-famous known series of books which were used in French universities from the first two years, called "Treaty of Special Mathematics", by Cagnac, Rammis and Commeau.

I've only read Tome 1 so far, but I know the table of contents of the rest. It was published in the 1970 and it shows.

The structure is as follows :
>Tome 1 : Algebra
Begins with Proofs, rudiments of set theory, (constructing integers/negative numbers/relative numbers), elements of Group Theory, elements of Combinatorics, Vectors (and vector spaces), complex numbers and the geometry thereof, finishing with matrices and polynomials

lipn.univ-paris13.fr/~duchamp/Books&more/Ramis/CRC/Cagnac-Ramis-Commeau- I -Traite-de-mathematiques-speciales-tome1-Algebre.pdf

>Tome 2 : Analysis
Basically ALL of calculus (from the very start !), suits and series, convergences, some bases of topology, some real and complex analysis, diff eqs. Also, construction of the real numbers.

lipn.univ-paris13.fr/~duchamp/Books&more/Ramis/CRC/Cagnac-Ramis-Commeau-2 -Traite-de-mathematiques-speciales-tome-2-Analyse.pdf

>Tome 3 : Geometry
Mostly outdated stuff, Euclid's elements, conics. Some nice stuff in Algebra, though.

Couldn't find it online.

>Tome 4 : Applications of Analysis to Geometry
Curves, double, triple, curve, surface and flux integrals, vector analysis, bases of differential geometry. Also, some mechanics at the end.

lipn.univ-paris13.fr/~duchamp/Books&more/Ramis/CRC/Cagnac-Ramis-Commeau-4 -Traite-de-mathematiques-speciales-tome-4-applications-de-l-analyse-a-la-geometrie.pdf

Keep in mind that :
Those are in French
Those are very old (1970), and some definitions are outdated (he defines eigenvalues from the trace of a matrix)
While they assume almost no knowledge of mathematics, they were intended for people with already a very good level of Pre-Calc, trigonometry, and the like.

Those books were written basically as an intro for the Nicolas Bourbaki books. As such, the problems in it are rather hard.

At the time, French mathematicians loved geometry and 50% of the problems are geometric in nature.

Not the maker of the list. But 2-3 people so far have praised what's in it. You have no business calling it a meme list when you advertise fucking RUDIN for the beginner like a fucking sperglord.

>Not the maker of the list. But 2-3 people so far have praised what's in it.
why do you keep lying? the only positive post is
>Good books in this list

> You have no business calling it a meme list when you advertise fucking RUDIN for the beginner like a fucking sperglord.
also you have me confused for someone else.

>why do you keep lying? the only positive post is

I haven't lied whatsoever.

Are not me, though I think two of the posts were made by the same person.

>also you have me confused for someone else.

Every single person who calls it a meme list is the kind of person who would, ironically advertise Baby or Papa Rudin as a person's first foray into mathematics. It makes no sense and it makes me believe that Veeky Forums is populated by people who want to see newcomers fail.

8966009 doesn't even mention your memelist
8966107 doesn't even mention your memelist
8965792 is whoever made the memelist
8965975 doesn't even mention your memelist
you really think these four posts are '2-3 people' praising the memelist?

>Every single person who calls it a meme list is the kind of person who would, ironically advertise Baby or Papa Rudin as a person's first foray into mathematics.
i just said that wasn't me, why are you so confused?

>It makes no sense and it makes me believe that Veeky Forums is populated by people who want to see newcomers fail.
the only people setting up newcomers to fail are the people posting memelists that will get newcomers absolutely nowhere

Not interested in playing the "ignore context" game with you. Those posts are either responding to criticisms of the "meme list" or they're referencing books from the meme list.

>i just said that wasn't me, why are you so confused?

If it looks like a duck and quacks like a duck...

>the only people setting up newcomers to fail are the people posting memelists that will get newcomers absolutely nowhere

What is wrong with the list? It seems perfectly reasonable to me. The only meme lists are the ones that encourages everybody to jump into Rudin.

> Those posts are either responding to criticisms of the "meme list" or they're referencing books from the meme list.

>Everything is a meme
>Just dive in nicca
really? what criticism is this responding to? what book is this referencing?

>If it looks like a duck and quacks like a duck...
Not interested in playing the "ignore context" game with you.

>The only meme lists are the ones that encourages everybody to jump into Rudin.
wrong.
every list in this thread so far is a memelist

Don't get angry at the trolls and autists, they know not what they do. There exist a few lists on Veeky Forums are truly meme-lists and only made to troll people with. As such there exists a certain hatred here for lists in general because there are many bad ones and the good ones get criticized along with the bad ones in the same light. Also, we have a lot of autists here who, while they may have learned the subjects, have no clue how to introduce it to someone else and how to guide them into the subjects. It's the "idk just do it" mentality that even professors do from time to time and they forget that sometimes people need to be guided because, guess what, they haven't learned to subject yet.

>Pre-Calculus - Carl Stitz & Jeff Zeager

Can be also be covered by Axler;s Precalculus. Distance 3rd and 4th is Lang's Basic Mathematics and Precalculus In A Nutshell.

>Linear Algebra and Its Applications - David C. Lay

Hard to go wrong with any of the modern linear algebra texts out there. Strang, Lay etc. What matters is that eventually if you go into pure math that you eventually cover linear algebra again rigorously through Axler and/or Kunze.

>How to Think Like a Mathematician - Kevin Houston
>How to Prove It - D. J. Velleman

Might be redundant.

>Calculus Vol. I & II - T. M. Apostol

Training wheels might not be necessary at this point. Also, consider adding a rigorous linear algebra text in addition to or in place of an Apostol-like book on calculus.

--

Otherwise, not bad. I rate 7/10.

Browse MAA Reviews.

Ther ratings are

BLL*** = essential as Apostol & Spivak
BLL** = strongly recommended as Courant
BLL* = recommended as Lax's Calculus
BLL = sugested as Velleman's Calculus

Calculus:
maa.org/press/maa-reviews/browse?field_tags_tid=36923

For Analysis I recommend the praised & new book series written by the Caltech Professor Barry Simon
Real Analysis: A Comprehensive Course in Analysis
maa.org/press/maa-reviews/real-analysis-a-comprehensive-course-in-analysis-part-1
With 5 volumes.

Spivak doesn't teach optimization or related rates among other techniques. It is not a calculus book, it is an intro analysis book.

Clark's book certainly seems to be an improvement over Spivak for many reasons. It gives an actual axiomatic system for the real numbers (Spivak just says something vague about you having encountered the reals before and then lists a few useful properties). It also builds intuition by providing an informal introduction to certain topics. As a result the formal sections later on are better than Spivak's treatment, in my opinion. There are many things I would do differently and I think the book suffers from its attempt at being a freshman introduction to analysis. I really don't like it's intro to proofs chapter (Mathematical Induction), it wastes time on things that in my opinion shouldn't be covered there and leaves out a lot of introductory proof concepts. As a result it feels disorganized and I'm honestly not sure that the intended audience (freshman math students) get much out of it.

>Velleman
Only objectively good recommendation on that list.

>Apostol
Teaches integration before differentation. Overpriced. Just kinda weird.

>Terrance Tao
Far too monolithic. Doesn't seem care about axiomatic systems and models, instead just stirring everything into one big pot. Good for learning that area of math, bad for learning abstractions.

cont.

There are many books better than Spivak but I haven't found a 'perfect' book yet. There are some great lecture notes out there as well but lecture notes tend to be less consistent in quality.

This user recommended Clark and provided a link. It seems to be a pretty reasonable book, and available online as well.
Clark's book is also very comparable to
>Steven R. Lay's Analysis with an Introduction to Proof Though Lay's book has a better (but still flawed) intro to proofs chapter it has much weaker motivation and intuition. Lay's book is at times more formal and at other times less formal than Clark's (though like Clark it begins with an axiomatic system for the reals). Otherwise they're fairly similar.

Here is a beautiful quick introduction to topology.
dpmms.cam.ac.uk/~twk/Top.pdf
Having read this a lot of the definitions, theorems, and proofs in the analysis texts can be made easier (both to perform and to digest).

Ideally an analysis text would have or provide an introduction to formal proofs and some basic introduction to topology. Then it would cover material formally (with an informal introduction for major concepts) using concepts from topology. I belive that if done correctly the book could be much shorter (even with an intro to proofs and a babby intro to topology) and easier to digest.

Many people taking analysis courses struggle to do basic problems and proofs because many of the sentences they're proving contain upwards of four nested quantifiers and they're never explicitly taught how to organize or approach a proof like that. So instead they end up doing weird shit where they work pieces of the proof out and then work their way backwards to a normal proof. Many of these people then have poor intuition for why this constitutes a proof and only do it this way because it's how they are taught. A proper understanding of formal proofs trivializes all of this wasted effort away.

Clark's book is also very comparable to
>Steven R. Lay's Analysis with an Introduction to Proof
Though Lay's book has a better (but still flawed) intro to proofs chapter it has much weaker motivation and intuition. Lay's book is at times more formal and at other times less formal than Clark's (though like Clark it begins with an axiomatic system for the reals). Otherwise they're fairly similar.

fixed

I have no idea if this is written by the same person, nor what you're trying to say (if you are the same person) thanks to the confusing order. Is Clark a worthwhile improvement on Spivak or is it underwhelming to a freshman math student? Is an undergraduate introduction to topology the best introduction to advanced mathematics in that it trains students how to go from basic to more complicated proofs? Do your suggestions incorporate this understand or is this left to the autodidact?

>Is Clark a worthwhile improvement on Spivak
Yes, definitely.
>or is it underwhelming to a freshman math student?
It is written for freshman math students which means the author is not as formal or terse as they could be (a lot of time is wasted covering material a student should ideally pick up in other freshman math courses). At times doing things in strange ways. For instance, they give an informal overview on limits and continuity and then give a formal definition of continuity and use it to define limits (rather than using limits to define continuity or just defining each separately).

>Is an undergraduate introduction to topology....complicated proofs?
No. What I meant was that topology introduces a notion of a topological space as well as a more general definition for continuous functions (as special sorts of functions between topological spaces). If someone has this basic background then they can understand the reals as a simple topological space and then many of the difficult to understand definitions and theorems regarding continuous functions become much easier to grasp. That said it is not necessary for a student to have a full blown introduction to topology in order to benefit from the abstract machinery of topology. Consider how Clark's book begins by giving the field axioms and uses them throughout the text without expecting the student to have any background in abstract algebra (simply put the majority of abstract algebra is not necessary background, and similarly neither is the majority of topology). An ideal analysis text would give a brief intro to topology.

On an unrelated topic: A basic background on theorem proving could be given at the beginning of the text but ideally one should instead have a freshman intro to proofs course and/or intro logic course as a pre-requisite (not topology).

Do your ... autodidact?
Mostly left to the auto-didact. I am not aware of a perfect book but there are certainly many books better than Spivak.

Me again,
I imagine there are a large number of informal math students who are confused about my comments on formality and proofs. Here is an example of what I would consider a good basic explanation of theorem proving for analysis (specifically with how to deal with nested quantifiers).

math.wisc.edu/~robbin/521dir/cont.pdf

A student who has taken a good introductory proofs course or an introductory formal logic course would have already known how to:
>Write these sentences formally.
>Negate the sentence if necessary (if one is attempting to "prove the original sentence is false").
>Structure and fill in a proof of the sentence.

I second this, I'm reading through it right now and it's fantastic, they also have interludes with historical content that actually enhances the experience and makes for a very fun read.

His proof of the chain rule is a crime against humanity.

Why? It's pretty standard.

Does anyone like Larson?

burn it

Why? What is bad about it?

It's a worse version of Stewart. And his other books and typical school books that are replaced with better books such as the AoPS series or Gelfand.

Here's a rule of thumb: if it has more than 4 or 5 editions, don't use it.

i see, thanks user

Since you guys are debating books

I am wrapping up an intro to proofs class next week and we used Mathematical Reasoning by Eccles.

Next quarter I am taking real analysis and we are doing the first 4 chapters of Rudin. I don't feel ready. What should I read over the fall? I already have Rudin in my possession if the best use of my time would just be previewing the book.

I heard Spivak is good to do before Rudin however.

I can read and write proofs but I am still a beginner having only learned how to do so about 9 weeks ago, so I'm not sure

I'm not ready for anal you haven't even taken me out for dinner yet.

Abbott and/or Alcock

>I heard Spivak is good to do before Rudin however.

There are fields of math outside of calculus and analysis you know. Rather than doing an "advanced calculus" book before analysis, go learn some other field of math like linear algebra or topology and improve your mathematical reasoning skills there. You'll be prepared for Rudin [math] and [/math] you'll know more math. Pick one of these and read them:

amazon.com/Principles-Topology-Dover-Books-Mathematics/dp/0486801543/
amazon.com/Combinatorial-Introduction-Topology-Dover-Mathematics/dp/0486679667/
amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514
amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/
amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407/
amazon.com/Combinatorics-Graph-Theory-Undergraduate-Mathematics/dp/0387797106/

I've already studied linear algebra, and I'll be taking abstract algebra in the fall along wit probability

The real analysis class is notorious for weeding out would be math majors and since I have an interest in graduate school, I want to do well in the course - which is why I want to prepare specifically for it

Then study point-set topology. That's what chapter 2 of Rudin is all about and what many people struggle with early on.

bump

Dont read, do exercises fucking brainlets, no wonder why you all suck and, as a jew, I hate you so much

The Calculus 7/TC7 by Louis Leithold

how do you do exercises effectively in a timely manner?

by doing them instead of posting on Veeky Forums about doing them

The real list:
>Step 0
Assess your skills, define your goal
>Step 1
Evaluate your options, either pick the book that looked most fun/ interesting and relevent to you or request a specific recommendation in person or online
>Step 2
Git gud

That was always in this list: If your math skills are way beyond that list, then you shouldn't be on Veeky Forums anyway. If you can hardly count, then go to Khan Academy and get your GED before even thinking about learning mathematics.

If you aren't majoring in math and instead are majoring in the sciences or engineering here is my suggested list of books in the order I went through them before i had to take the course for them, buy them in paper for quick reference.

>Book of Proof by Richard Hammack
Proofs become extremely important later on in math especially during PDE/Complex Analysis so do them ASAP whenever you have free time.

>Calculus by Stewart
This is the standard and it's the standard for a good reason.

>Ordinary Differential Equations by Tenenbaum & Pollard
Start this ASAP, has 65 lessons in it, which means you can finish it in 65 days, specifically 5 or 6 lessons take a day or two due to the # or complexity of the problems, the rest you can finish under a day.

>Matrices and Linear Algebra by Hans Schneider
This will suffice if your aren't a math major, covers the standard curriculum of LA with LOTS of matrix calculations, extremely first time beginner/intro friendly, you'll be prepared for LA if you do this book entirely.

>Matrices and Linear Transformations by Charles G. Cullen
Just like the previous but more in depth, you'll be fine for LA if you only did Schneider though.

>Applied Partial Differential Equations by J. David Logan
Short, concise, great, never skips a step, if you go through this before you're first course in PDE's you'll be good for it.

>A First Course in Complex Analysis 1.5 Edition by Matthias Beck
>An Introduction to Complex Analysis 2011th Edition by Ravi P. Agarwal
Both are first time friendly, I did both at the same time, really great, got an A in the course due to doing both of these before the semester started, simple and clean, never skips a step.

>Introduction to Partial Differential Equations by Peter J. Olver
Solutions posted online, if you're using Strauss's PDE book go through Olver's first because it's better, this book never skips a step in proving something and it has the solutions. Strauss skips steps and has doesn't have all solutions.

>Book of Proof by Richard Hammack
Good stuff.

>Calculus by Stewart
It's the standard because nobody knows about Knisley yet. It's a good book for problems but it's terrible for developing intuition outside of rote memorization and drilling.

>Ordinary Differential Equations by Tenenbaum & Pollard
Classic. Can't go wrong there. But I think Vladimir Dobrushkin is better.

>bad lin. alg. textbooks
Just go with David Lay or Gilbert Strang for your introduction to linear algebra.

>A First Course in Complex Analysis 1.5 Edition by Matthias Beck
What do you think of Needham?

>Applied Partial Differential Equations by J. David Logan
Thank you for bringing this to my attention.

I think it doesn't really matter what goddamn textbook you will read.
U can read 2 or 3 simultaneously. Major thing here is how much knowledge u can get, not "read the coolest meme book haha"

Nobody wants to read a shit textbook that's littered with typos, doesn't explain anything, or doesn't even have solutions.

You're wrong because textbooks do matter, some are shit and some are great.
This

Solutions? Haha, i think they on some degree curruptive and teach to go easy instead of hardworking.

>teaches
yare yare poor english

>bwahaha who needs solutions
Might as well just read jackson's electrodynamics, who cares about steps if we don't care about solutions, you're the type of bad influence that would recommend apostal as a first time calculus book and have them start off with griffiths E&M

Whatever, do what u want, im just sharing this hella generic opinion.
But there is no ready-made solutions in real science...

>Gilbert Strang
Is it really that good? Like if I have never seen LA and i self studied that i would have no trouble like the other LA books?

>What do you think of Needham?
Pirated it, didn't like, doesn't help that i failed geometry twice back in HS, so looking at that book made me more confused than anything, i liked Beck because he just went straight forward into explaining everything as simple and concise as possible

I don't care if the book doesn't have solutions if I'm taking a course at a university. Then I can go to office hours if I'm truly stumped after trying my hardest. But if I'm self-studying? I'm not going to spend ages trying to solve a problem, especially if it turns out to be totally flawed in construction. I don't have a thousand years to live you insufferable faggot.

Nobody reads a textbook when performing real science. They read a textbook to prepare for real science. And that means learning the basics so you can apply it to the world. Stop being an autistic cunt.

Are u the autistic one? Can't you ask for help? Even solution can be too hard for understand.
And you are like obsessive broken record. Why can't you approve my right to have different opinion?

>Are u the autistic one? Can't you ask for help? Even solution can be too hard for understand.
Not when you're not in university you fucking retard. What am I going to do? Rely on you shitheads? I'd be setting myself up for failure.

>And you are like obsessive broken record. Why can't you approve my right to have different opinion?
I never said you didn't have the right to express your shitting opinion. Why can't you approve my right to criticize your opinion?

You are autistic and cannot convey your words as you want to convey them.

Wow. Of course u have. And go out with a cool math nerd, i hope you and he will be fucken happy to discuss some problems.

How do I even go about reading shit like Spivak while taking courses that require different textbooks?

no it wasn't

what are the Veeky Forums approved textbooks for intro to electricity & magnetism?

if you know cal 3 then purcell E&M, which is usually an honors course for E&M
but it's based on special relativity, so youd had to read his SR textbook before diving into E&M

It goes in this order

Special Relativity: For the Enthusiastic Beginner Paperback – January 20, 2017
by David J Morin

Electricity and Magnetism 3rd Edition
by Edward M. Purcell & David J. Morin

Veeky Forums-science.wikia.com/wiki/Physics_Textbook_Recommendations#Electrodynamics

What do you think about Strang's Calculus? Isn't it good enough for self-studying in comparison to "meme books"?

Just when you think you've find a good list people yet again shit all over it. What the fuck

You're always going to find autistic braindead retards shitting on intro lists because they don't start people with hardcore real analysis. There's a lot of good recs in here. Don't worry about it.

>that retard suggesting learning multivariable calculus before linear algebra
you can't take anything he says seriously

You don't.

I never understood the multiple textbooks for one class meme. This only makes sense when you're doing higher level stuff that has several layers of abstraction and you need a different explanation to understand something.

But working through two calculus books at once while taking a course in calculus is dumb. Just focus on your school work and the assigned textbook. Do the exercises and get an A

Don't bother with Spivak at all. Just read your assigned textbook. Spivak is only for confirmed mathematicians. It's abstraction and rigor is useless for anyone else.
Griffiths is a good intro. Later Jackson but you'll probably never need this level of EM.
Comparable to Stewart. It's solid. I recommend reading Calculus Made Easy by Thompson first before any calc book.
That many books before analysis? It really doesn't require that much preparation. Just start reading Spivak after a proofs book and it should be fine.
Read Rosenlicht's analysis. It's much easier than Rudin and being familiar with the concepts will help a ton. You only have to read up through "interchange of limit operations", the rest is optional. You can do it in a month since that's longer than how long it took me.
Here's my list for math majors or theoretical physics.
Calculus Made Easy by Thompson.
Learn basic calc, lin alg, diff eq, multivariable. Stewart, Lay, Boyce and DiPrima are what I used but anything is fine.
Linear algebra from Axler and Hoffman and Kunze. Take these seriously as it'll be your first foray into rigorous math. Look up mathematical induction.
Analysis from Rosenlicht. After that look at Rudin's principles, or you could leave it for later, after you've read Spivak's Calculus on Manifolds.
Algebra from Herstein(Topics in algebra). Lang's undergraduate algebra is a nice quick read. Many people love Artin and Dummit and Foote so those are worth a look.
Topology. Munkres is the standard. But all of them seem to be fine to me.

Courant > Spivak

What do people think of basic mathematics by Lang as a first book?

As a first book do mean to someone who has never learned algebra? Not a good idea. Basic mathematics is intended to review and cover everything one needs before entering a calculus course. It's a good textbook but not suited for a first timer. A better idea would be the Art of Problem Solving books if you have a decent tutor or teacher helping you.

How about when you learned some algebra in high school more then 4 years ago? What would you recommend for self study without a tutor?

If that's the case, go look up some youtube videos to remember a few things and Google a pdf for Basic Mathematics. Also consider looking into this stitz-zeager.com/szprecalculus07042013.pdf and see which one you like better.