Kindly refer me the best book for the above mentioned topics. Actually, I'm looking for both theory and problems and it will be good if the book handles the topics in depth.
I didn't find any single book which covers all the topics... so can you kindly mention different books for different topics
Two books immediately come to mind, both by the same authors Mac Lane and Birkhoff's "Survey of Modern Algebra" and Mac Lane and Birkhoff's "Algebra" The latter is more advanced than the former and is more comprehensive. I am a big fan of Mac Lane's mathematical style: rigorous and interesting.
For all the things mentioned in your 1st. para 'Theory of Equations' the best book is called, er, 'The Theory of Equations'. By Burnside and Panton. Just vol 1. Vol. 2 is devilish, but vol 1 has all the things you mentioned. I think maybe Panton wrote the first vol and Burnside the second. Vol 1 of the Theory of Equations is the best book ever written on the theory of equations. In fact it is the best book ever written. So later let me know how you got on with it. If you have found it helpful I might even read it myself.
For the inequalities and set theory topics, I think Wikipedia is sufficient. Post a question if there's a detail you don't understand. There's a lot of linear algebra on your list. I like Axler's book, but you might prefer one that introduces determinants earlier. (I don't think you would have any problems with Axler, but if you want to you can find lots of other recommendations by searching this forum).
Thanks for ur replies :) However,the books mentioned in the above posts doesn't deals with "Theory of equations" particularly Newton's identities, Reciprocal equations, Rolle's theorem, Des cartes and rule of signs. Is there any book which covers these topics or shuld i read these topics from internet??
Burnside and Panton does deal with every one of those. When I imply I haven't read it I am joking of course. I have never read anything else. It is old (7th edn. 1912! repr. Dover, you should be able to find it) and 'traditional' rather 'concrete'. There may be shorter books on the same subject - that by no means necessarily makes them easier! Let us hear other opinions to choose what best suits you.
Artin (for algebra) and the first few chapters of Baby Rudin (for analysis) will have you covered on all topics, and in plenty of depth (Baby Rudin especially) http://www.amazon.com/Algebra-Micha...=sr_1_1?ie=UTF8&s=books&qid=1266059414&sr=8-1 http://www.amazon.com/Principles-Ma...=sr_1_1?ie=UTF8&s=books&qid=1266059439&sr=8-1 Baby Rudin (aka Principles of Mathematical Analysis) isn't for the faint of heart though. It's written in an extremely concise style relative to lower division and most upper division undergrad books, and you'll have to fill details like drawing diagrams and filling in steps as you read it or you'll be totally lost (unless you have an IQ of 5000). Make no mistake though; there is an incredible amount of information contained in it. I honestly think the only way to read it is to sketch the proofs in full detail as you go along, because Rudin's arguments are really slick and can look like hand-waving if you don't get down and dirty with them. Artin's a lot more forgiving, but no walk in the park either. Both texts have very challenging problems, though Artin has quite a few of the plug and chug variety while Baby Rudin has very few. If I had to make one recommendation though, have your multivariable calc class down cold. Or if you're not in school, make sure you have all the topics from this multivariable class below down cold: http://ocw.mit.edu/OcwWeb/Mathematics/18-02Fall-2007/CourseHome/index.htm You gotta know your geometry or you'll just be doing symbol manipulation. As an aside, another plus to reading Artin is you'll be able to watch these lectures from a Harvard algebra class as you go along with it. http://www.extension.harvard.edu/openlearning/math222/
If you go with the Baby Rudin book for your analysis studies, you can also follow this class below (there's also problem sets and solutions) http://ocw.mit.edu/OcwWeb/Mathematics/18-100BFall-2006/CourseHome/index.htm
@epenguin, Thanks :) it covers most of my linear algebra part. @gauss^2, Thanks for the video links and i think for abstract algebra i will go for artin :)
You know I started off reading the 2nd edition of baby Rudin (I switched to Pugh, a much better book), and later got the 3rd edition. Believe it or not, the 2nd edition is even more difficult that the 3rd! The third edition helps you more in the questions, proofs and examples, and has more easier questions (easier than the others he has, but by no means easy in absolute terms). The presentation is also cleaner in the 3rd edition, in my opinion. Just some trivia for you. But yea, I think Pugh's Real Mathematical Analysis is by far the better book.