Exploring Analysis: Suggested Books for High School Students

[polarbears]

Hi, I'm a high school student that has completed my most of my under division in mathematics (Diff. Eq., Discrete math, single/multi variable calculus, linear alg, problem solving, and basic group theory stuff) and I'm now interested in Analysis.

Can someone suggest an introductory book to this subject?

 

[JG89]

If you haven't yet learned theoretical single-variable calculus, the following three books are best to study from:

"Introduction to Calculus and Analysis Volume 1" - Richard Courant & Fritz John

"Calculus" - Michael Spivak

"Calculus" - Tom Apostol

If you have already learned this, you may want to pick up Rudin's "Principles of Mathematical Analysis", Apostol's "Mathematical Analysis" or Charles Pugh's "Real Mathematical Analysis"

 

[lugita15]

I read a lot of Introductory Real Analysis by A.N. Kolmogorov in middle school. Although there was quite a lot I didn't understand at a time, it was definitely a fantastic introduction. With all of your math background, I think you'll definitely be prepared for it.

 

[Studiot]

Either
'Calculus'
or
'Mathematical Analysis'

Both books by Binmore are modern and first rate.

 

[HallsofIvy]

Calculus and Differential Equations, as taught in high school is seldom at the same leve as the same course taught in college- and tend to vary from high school to high school or even teacher to teacher far more than college courses. That makes it difficult to say whether you should be looking at a (theoretical) calculus text or an analysis text.

 

[snipez90]

Introductory Real Analysis by Kolmogorov and Fomin is the most advanced of any text mentioned, so this is probably not a good place to start. There are two chapters discussing metric spaces and topological spaces, and then about 3 or 4 chapters on functional analysis, and finally three chapters on measure theory and integration. Even if you could follow many of the arguments in the text, you would still be missing out on a lot of basic real analysis, which is more elementary but fundamental. You won't get very far in Kolmogorov and Fomin if you're not well-versed in epsilon-delta arguments.

If you want a cheap intro analysis text that is fairly excellent, I would recommend Maxwell Rosenlicht's Introduction to Analysis. This text is very easy to read, and is probably a good supplement to a more comprehensive text such as Apostol's analysis text. It starts with the axioms of the real numbers and culminates with a discussion of analysis in R^n. Since you can get the Dover copy for like 10 bucks, it's also a good deal.