Real and Complex Analysis Textbook

[Reedeegi]

I'm currently looking for a textbook on Real and Complex Analysis. I currently own both Rudin's and Shilov's, and I'm interested to know if there are any more with that scope of topics. In English, please.

 

[marcusl]

Whittaker and Watson, though old, is considered the golden classic.

 

[PhDP]

...I'm looking for an introduction to real analysis, is Rudin's book appropriate (I've many bad things about it, especially from physicists) ? I'm a scientist with 7 courses in maths (Calculus I/II, Multivariate Calculus, Ordinary Differential Equations, Linear Algebra I/II, Statistics).

 

[profanilp]

rudins book is worse book evr written (realasnd complex analysis)

 

[statdad]

...I'm looking for an introduction to real analysis, is Rudin's book appropriate (I've many bad things about it, especially from physicists) ? I'm a scientist with 7 courses in maths (Calculus I/II, Multivariate Calculus, Ordinary Differential Equations, Linear Algebra I/II, Statistics).

Not sure why rudin's books are considered poor by physicists: they are math texts and should not be expected to cater to people in other disciplines. I used "baby" rudin (my edition was blue) as a senior for introductory analysis and as a first semester grad independent study for multivariate analysis/basic measure theory. found it very easy to read with a good selection of problems. I used the other rudin as a reference through grad programs.

 

[lavinia]

I'm currently looking for a textbook on Real and Complex Analysis. I currently own both Rudin's and Shilov's, and I'm interested to know if there are any more with that scope of topics. In English, please.

Royden's book on Real analysis. Much easier than Rudin which is too hard for most mathematicians.

Alfors book on Complex Analysis - a classic but not inspirational.

Visual Complex Analysis
by Needham

Books on stochastic processes teach some of real analysis and give specific meaning to many of the ideas. A First Cousrse in Stochastic Processes Ksrlin and Taylor also a second course

 

[statdad]

Royden is also good.

This
"Much easier than Rudin which is too hard for most mathematicians." is ridiculous.

 

[Gib Z]

People who say Baby Rudin is too hard for mathematicians are probably misunderstanding the purpose of his book. By no means is it intended to be a readers first yet complete course in Real Analysis, Topology, Differential Forms, Lebesgue Theory and more, all in a few hundred pages. It's for people who have already taken a course or two in each of those topics, and are honing in their skills in them. If you use the book it was intended to be used, it is no doubt a valuable resource.

 

[profanilp]

baby rudin is terse! and not a good book.
reasons: riemann integral is defined for realvalud functions and definition heavily relies on order properties of R. With very less efort the whole theory works in banach space setting see Dieuodenne or lang analysis I
treastment of stokes theorem is too hurried and turn to spivak or lang analysis i or Ii.
the authour discuees abstarct maths but uses old fashioned term quantity.
rather than implicit function theorem inverse function theorem is taken as a strting theorem . see dieuodenne foundations of modern anlysis
bartle or apostol are better replacements and lang anlysis I is highly recommended .
just like a mad fashion people read rudin which is neither lucid not completely .
goldberg is excellent for one variable.
pugh real nalysis is also very nice new arrival.

 

[statdad]

"baby rudin is terse! and not a good book.
reasons: riemann integral is defined for real valud functions and definition heavily relies on order properties of R. With very less efort the whole theory works in banach space setting see Dieuodenne or lang analysis I"
That is not the intent of rudin's text.

"treastment of stokes theorem is too hurried and turn to spivak or lang analysis i or Ii."
Not sure what you mean: not enough coverage or too many pieces for the reader to turn in.

"the authour discuees abstarct maths but uses old fashioned term quantity."
You do know this text was available in the late 70s early 80s, correct? how old are you referring to?

"rather than implicit function theorem inverse function theorem is taken as a strting theorem . see dieuodenne foundations of modern anlysis"
So - not a problem.

bartle or apostol are better replacements and lang anlysis I is highly recommended ."
Saying better here is a personal judgement. I'm not a fan of lang's writing: bartle and apostol are both good.

"just like a mad fashion people read rudin which is neither lucid not completely ."
This is a statement completely lacking in substance. Are you implying that it is incorrect for an author to demand a little thought and work from his readers? I would rather not be spoon fed.

"goldberg is excellent for one variable.
pugh real nalysis is also very nice new arrival."
I'm not familiar with either of them.

Again: I'm not sure why there is this rabid dislike of rudin, but if you take the time to actually read what I've written neither have I said it is the best introduction to real analysis text in the history of the universe. My strongest comment was against the statement to the effect that "rudin was the worst ever" - unless the person who wrote that has reviewed every text ever written, it is impossible to justify that statement as being one of hyperbole. As your comments seem to be.

 

[profanilp]

Soory . rudin's principles of math nalysis is not a good text not beacuse it is terse. but
1) It offers atreatemtn of riemann integral heavily fdepndent on order and unsuitanle gfor generalization to vector valued maps. se bartle excellent
2) it does not offer good treatment of riemann integral in several variables .see for example lang( anlysi I or bartle ) .
3) on the contray it rather hurriedly offers stokes theorem ( a better version is avlaible in spivak or anlysi I by alng or analysis Ii by lalng)
4) rudin's ral and complex analysis is not good beacuse it does neglect series developments an integral part of complex . and does not discuss laurent series!
2) it offers a complicated algorithm to find winding number which does not accomlish much!
3) very childishly it offers mesuarble sets and mappings in parallel with contiouous amppings and open sets where the entir motivation of these two theories differ except for syntactic similarities.
4) the constructionof measure is not nicely doen .e ither follow pure daniell approach as in apslung bungart or follow lang as in analysis II
5) the author beleives in assembling amny theorems . as u can see many of ther esults selected for presentation are not relevant today. see in contrast lang analysis Ii . u learn from this book then u are at the frontiers of real nalysis today. lang or alhfors complex really give crux of complex .
6) quantity is term use in 30s not in seveties and eighties. Rudin was a student of racist professor moore!
6) functional nalysis , function theory in polydisks, Fourier analysis on groups are good books by rudin.

 

[snipez90]

Okay I won't bother responding to the complaints regarding several variable theory since I didn't study that from Rudin. Responding respectively,

4) Rudin does not neglect series developments. Also, Laurent series are actually left to the exercises since it probably detracts from his particular treatment of basic complex variables in a single chapter.
2) What complicated algorithm? I'm pretty sure he just calls the winding number the index and then uses it to give mostly standard proofs. Not sure what your complaint is here.
3) Eh, I felt the analogy with basic topology helped in getting down the basics of measurable sets and measurable functions. Obviously general topology and analysis begin to differ when you start talking about say countability axioms and metrization problems, but the similarities between basic facts regarding continuous functions and those of measurable functions are apparent to a lot of people, including myself.
4) Please explain why it is not nicely done in Rudin instead of suggesting alternative constructions. Why should I follow a daniell approach?
5) This is somewhat of a legit complaint. Yes Rudin does not always give context for the many theorems he puts together, but this is exactly why it also makes a good reference. I don't think his presentation really affects his clarity and exposition skills though.
6) You're thinking of his wife, who studied under Moore. Walter Rudin studied under John Jay Gergen at Duke. In any case, does studying math under a racist make you a racist?
6) Eh I disagree, but then again I mostly stayed away from his texts while studying functional and Fourier on LCA groups.

Anyways, the advantages of Rudin are myriad, the disadvantages are trivial, and further argument will lead to loss of valuable analysis time.

 

[lavinia]

Royden is also good.

This
"Much easier than Rudin which is too hard for most mathematicians." is ridiculous.

Would you like to discuss that?

 

[statdad]

Would you like to discuss that?

Very simply: yes. Defend the "too difficult for most mathematicians". On what basis does that make sense?

Are there people who don't like Rudin? Yes - for some legitimate reasons, no doubt, some (as advertised above) simply because it doesn't fit what the reader thinks appropriate.

I don't think textbooks should spoon-feed the intended audience. If you aren't willing to work along don't complain about the writing being "terse" - and stay away from statements, like the above, that are pure hyperbole (unless, of course, you have surveyed "most mathematicians" and documented the results).

 

[lavinia]

Very simply: yes. Defend the "too difficult for most mathematicians". On what basis does that make sense?

Are there people who don't like Rudin? Yes - for some legitimate reasons, no doubt, some (as advertised above) simply because it doesn't fit what the reader thinks appropriate.

I don't think textbooks should spoon-feed the intended audience. If you aren't willing to work along don't complain about the writing being "terse" - and stay away from statements, like the above, that are pure hyperbole (unless, of course, you have surveyed "most mathematicians" and documented the results).

Some textbooks are so complicated and so abstract that working mathematicians avoid them just because the level of difficulty is unnecessary for them to do their research. this is what I meant by too hard. Another classic example of such a book is Spanier's Algebraic Topology. Both Rudin and Spanier are elegent and wonderful - but in practice they are just too hard (in my opinion. And Rudin is way to hard for a beginner in analysis.) I could go into detail on both books what it is about them - but you probably think I am arrogant and preposterous so I won't annoy you further.

 

[xienohp]

Rudin: Simple and detailly explained what you needed

 

[profanilp]

for measure theory royden a, lang and for complex lang , alhfors are more to the frontiers of the subject and are better.

 

[profanilp]

Rudin's book summartizes cauch theory one chapter but it is not at all a good acount as series expansions are neglected. Also the algorithm for winding number is difficult to figure out how it works . a simpler algorithm is presented in alhfors. butt looking at usefulness value( ? is there the algos are pretty trivial) it seems best not to have them. langs book rightly avoids the algorithm. peole commenting hre have not read text from line ti line. i read and found the algorithm not at all a proper one .
infact one can define inside of a closed curve using winding numbers ( not interior) and then cauchy's thm can be stated just as stated classically. one then does not require jordan curve theorem and one has intellectual satissfaction that i am not saying integral of f is zero when integral of 1/ ( z-a) is zero for all points a outside the region. actually it seems a good idea to define winding number in terms of uintehral of ( z-a) * conjugate so atht dependence on log or argument( indirectly) is avoided. no one has done that. in this sense. casuchy theory has not acheived perfection as yet even though dixon has brought it almost close to perfection.

 

[mathwonk]

i recommend cartan for complex and berberian for reals.

 

[lavinia]

Ok. For those who think that Rudin's Real and Complex Analysis is a good beginner's book I am going to post, for fun, some homework probelms at the end of his chapters - for our amusement.

First some easy ones.

- Show that any infinite sigma algebra is uncountable.

I once gave this one to a Ph.D. in probability theory and he was unaware of it - my point about this book - although he solved it in a couple of minutes.

More to come.

 

[lavinia]

well no solutions to the first one. But in hopes that all who find Rudin easy and straightforward here is a second.

Construct a Borel set so that the measure of its intersection with every closed interval is positive and strictly less that the measure of the segment. Can such a set have finite measure?

 

[lavinia]

A fact that has been mentioned recently in these threads is also a homework in Rudin. Show the the graph of a discontinous map from the reals to the reals that is a linear map over the rational numbers is dense in the plane.

This one actually is pretty easy but again a bit abstruse.

 

[Landau]

For what it's worth: I find Rudin's book very good; the exposition is beautiful and I often look up stuff in it. So I use it mainly as a reference. I don't know how good it serves as a "beginner's book". I haven't done any of the exercises, but scanning over them I see there are some darn hard ones.

Anyway, I always like to use several sources at the same time when learning a subject. As such I would recommend Rudin to be one of them, but probably not if it's your only source.

The following exercise is fun, but admittedly it took me quite some time; more than a couple of minutes!

First some easy ones.

- Show that any infinite sigma algebra is uncountable.

I once gave this one to a Ph.D. in probability theory and he was unaware of it - my point about this book - although he solved it in a couple of minutes.

More to come.

Let $$\Sigma$$ be a sigma-algebra on a set $$X$$. Define the following relation on $$X$$:

$$x\sim y:\Leftrightarrow (\forall A\in\Sigma: x\in A\Leftrightarrow y\in A)$$.

This is obviously an equivalence relation, basically because "iff" is. Hence we get a partition $$P=\{[x_i]\ |\ i\in I\}$$ of X. We have a nice decription of the equivalence classes, namely

$$[x]=\bigcap \{A\in\Sigma\ |\ x\in A\},$$

i.e. it is the smallest set contained in all things in our sigma-algebra that contain x. Indeed, because a sigma-algebra is closed under complements, we have:

(y is in that intersection) iff (for all $$A\in\Sigma$$ with $$x\in A$$ we have $$y\in A$$) iff (for all $$B\in\Sigma$$ with $$x\notin B$$ we have $$y\notin B$$).

So indeed

(y is in that intersection) iff (y ~ x).

Now suppose that $$\Sigma$$ is countable; we will show it must be finite. This means each [x] is in fact a countable intersection, hence $$[x]\in\Sigma$$. So we get the map $I\to \Sigma$ given by $i\mapsto [x_i]$; it is injective by definition of I (it indexes the partition). Hence I is also countable. But then we get a map
$$2^I\to \Sigma$$

$$J\mapsto \bigcup_{i\in J}[x_i]$$
(because such a union is countable by our previous remark, hence in the sigma-algebra). Again by definition of I it is injective. It is also surjective; in fact the inverse is given by
$$\{i\in I\ |\ b\in[x_i]\text{ for some }b\in B\}\mapsfrom B$$.

Hence $$2^{|I|}=|\Sigma|$$. If I were infinite then $\Sigma$ would be uncountable. Hence I is finite and consequently $\Sigma$ is finite.

 

[lavinia]

For what it's worth: I find Rudin's book very good; the exposition is beautiful and I often look up stuff in it. So I use it mainly as a reference. I don't know how good it serves as a "beginner's book". I haven't done any of the exercises, but scanning over them I see there are some darn hard ones.

Anyway, I always like to use several sources at the same time when learning a subject. As such I would recommend Rudin to be one of them, but probably not if it's your only source.

The following exercise is fun, but admittedly it took me quite some time; more than a couple of minutes!


Let $$\Sigma$$ be a sigma-algebra on a set $$X$$. Define the following relation on $$X$$:

$$x\sim y:\Leftrightarrow (\forall A\in\Sigma: x\in A\Leftrightarrow y\in A)$$.

This is obviously an equivalence relation, basically because "iff" is. Hence we get a partition $$P=\{[x_i]\ |\ i\in I\}$$ of X. We have a nice decription of the equivalence classes, namely

$$[x]=\bigcap \{A\in\Sigma\ |\ x\in A\},$$

i.e. it is the smallest set contained in all things in our sigma-algebra that contain x. Indeed, because a sigma-algebra is closed under complements, we have:

(y is in that intersection) iff (for all $$A\in\Sigma$$ with $$x\in A$$ we have $$y\in A$$) iff (for all $$B\in\Sigma$$ with $$x\notin B$$ we have $$y\notin B$$).

So indeed

(y is in that intersection) iff (y ~ x).

Now suppose that $$\Sigma$$ is countable; we will show it must be finite. This means each [x] is in fact a countable intersection, hence $$[x]\in\Sigma$$. So we get the map $I\to \Sigma$ given by $i\mapsto [x_i]$; it is injective by definition of I (it indexes the partition). Hence I is also countable. But then we get a map
$$2^I\to \Sigma$$

$$J\mapsto \bigcup_{i\in J}[x_i]$$
(because such a union is countable by our previous remark, hence in the sigma-algebra). Again by definition of I it is injective. It is also surjective; in fact the inverse is given by
$$\{i\in I\ |\ b\in[x_i]\text{ for some }b\in B\}\mapsfrom B$$.

Hence $$2^{|I|}=|\Sigma|$$. If I were infinite then $\Sigma$ would be uncountable. Hence I is finite and consequently $\Sigma$ is finite.

Nice. The equivalence class of a point may not be measurable if the sigma algebra is uncountable but your proof works by using the assumption that it is countable to get it to be a measurable set.

Another attempt would be to directly partition the sigma algebra into countably infinite disjoint measurable sets then repeat your union of subsets argument.

 

[Landau]

Nice. The equivalence class of a point may not be measurable if the sigma algebra is uncountable but your proof works by using the assumption that it is countable to get it to be a measurable set.

Yes, I believe I explicitly said that:

Now suppose that $$\Sigma$$ is countable (...) This means each [x] is in fact a countable intersection, hence $$[x]\in\Sigma$$.

I will think about your other two questions later :p
May I ask: what do you think about Rudin's book, besides it not being suitable for a beginner? Have you (been forced to) use(d) it in a course? Have you done many exercises? Do you use it as a reference?

 

[lavinia]

Yes, I believe I explicitly said that:I will think about your other two questions later :p
May I ask: what do you think about Rudin's book, besides it not being suitable for a beginner? Have you (been forced to) use(d) it in a course? Have you done many exercises? Do you use it as a reference?

I like to learn from elementary examples then generalize. Rudin goes the other way around. still the exposition is elegant and deceptively clear. The problems are often very hard but also very cool.

I have worked through a lot of the exercises but many I could not do.

I have never used it in a course. But studied it hard for a while. In the ways I have suggested I think it was a mistake.

 

[Skrew]

Note that "principles of mathematical analysis" is different then "real and complex analysis".

"Principles of mathematical analysis" information wise looked pretty understandable when I flipped through it but I think the exercise are going to be more difficult then books like krantzs(which I like).