# A good introductory book on real analysis with examples

## Main Question or Discussion Point

Hello

I am wonderinf if you guys know about a good introductiry book on real analysis where they use examples. I have found some books online, but they seem to not show with examples what we can do. I have read to Spivak calculus book is a good introduction, but I think it was too basic.

I need it to have basic set theory, basic measure theory, introduction to lebesque integration, and convergence of series.

Are there any books that got this, and got illustrations, and examples? What do you reccomend?

## Answers and Replies

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lurflurf
Homework Helper
I think it is taken to be the readers job to give examples. Do you just want a book that gives a lot of examples, one where giving examples is the main method of teaching, or a book of examples?

Two books which I think are worth looking at that may not be exactly what you want are
Counterexamples in Analysis (Dover Books on Mathematics) by Bernard R. Gelbaum
Counterexamples in Topology (Dover Books on Mathematics) by Lynn Arthur Steen
(as Topology needed in introductory analysis)

You might like Knapp "Basic Real Analysis."

WannabeNewton
You are being very vague with the term "introductory real analysis". A first course in real analysis will not usually cover measure spaces and Lebesgue integration in detail, at least not in the US. A first course in real analysis will be the usual analysis on ##\mathbb{R}##, metric spaces/normed spaces, and possibly some point-set topology. Usually a second or third course in real analysis will focus on measure theory (I say second or third because at some unis the second year is analysis on submanifolds in ##\mathbb{R}^{n}## and the third is measure theory and functional analysis, which is the case at MIT IIRC, whereas at other unis it is the opposite; of course some schools cram it all into one or one and a half years e.g. Harvard and UChicago).

But I think the real analysis book by Carothers (aptly titled "Real Analysis") will be perfect for you. It contains a summary/recap of analysis on ##\mathbb{R}## followed by a recap of set theory (cantor set, countability/uncountability etc.), goes into the topological and analytical concepts surrounding metric and normed spaces, then goes into function spaces (which takes care of your request for convergences of series), and finally goes into measure theory.

lurflurf
Homework Helper
^Well that is still the first course even if it is 2-3 terms. Many schools are organized by years instead of terms anyway. The courses are named confusingly as calculus and analysis are interchangeable. A first year honors/second year course may be called second year calculus, intermediate calculus, advanced calculus, or something analysis. A second year honors/third year/remedial graduate course may be called third year calculus, advanced calculus, or something analysis. The names and topics overlap.

lurflurf
Homework Helper
I think a book is Real and Functional Analysis (v. 142) by Serge Lang, It has many examples, but I do not know exactly what you want. If it is too bag a jump from Spivak try one or more of

Introduction to Analysis, by Maxwell Rosenlicht
Elementary Real and Complex Analysis, by Georgi E. Shilov
Undergraduate Analysis (Undergraduate Texts in Mathematics) by Serge Lang

Though they are probably a bit easy after Spivak.