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Reedeegi
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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.
PhDP said:...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).
Reedeegi said: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.
statdad said:Royden is also good.
This
"Much easier than Rudin which is too hard for most mathematicians." is ridiculous.
lavinia said:Would you like to discuss that?
statdad said: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 said: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.
Landau said: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 [tex]\Sigma[/tex] be a sigma-algebra on a set [tex]X[/tex]. Define the following relation on [tex]X[/tex]:
[tex]x\sim y:\Leftrightarrow (\forall A\in\Sigma: x\in A\Leftrightarrow y\in A)[/tex].
This is obviously an equivalence relation, basically because "iff" is. Hence we get a partition [tex]P=\{[x_i]\ |\ i\in I\}[/tex] of X. We have a nice decription of the equivalence classes, namely
[tex][x]=\bigcap \{A\in\Sigma\ |\ x\in A\},[/tex]
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 [tex]A\in\Sigma[/tex] with [tex]x\in A[/tex] we have [tex]y\in A[/tex]) iff (for all [tex]B\in\Sigma[/tex] with [tex]x\notin B[/tex] we have [tex]y\notin B[/tex]).
So indeed
(y is in that intersection) iff (y ~ x).
Now suppose that [tex]\Sigma[/tex] is countable; we will show it must be finite. This means each [x] is in fact a countable intersection, hence [tex][x]\in\Sigma[/tex]. So we get the map [itex]I\to \Sigma[/itex] given by [itex]i\mapsto [x_i][/itex]; it is injective by definition of I (it indexes the partition). Hence I is also countable. But then we get a map
[tex]2^I\to \Sigma[/tex]
[tex]J\mapsto \bigcup_{i\in J}[x_i][/tex]
(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
[tex]\{i\in I\ |\ b\in[x_i]\text{ for some }b\in B\}\mapsfrom B[/tex].
Hence [tex]2^{|I|}=|\Sigma|[/tex]. If I were infinite then [itex]\Sigma[/itex] would be uncountable. Hence I is finite and consequently [itex]\Sigma[/itex] is finite.
Yes, I believe I explicitly said that:lavinia said: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.
Now suppose that [tex]\Sigma[/tex] is countable (...) This means each [x] is in fact a countable intersection, hence [tex][x]\in\Sigma[/tex].
Landau said: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?
Real and complex analysis is a branch of mathematics that deals with the study of real and complex numbers and their properties. It involves the analysis of functions, sequences, and series in the real and complex domains.
This textbook is primarily intended for advanced undergraduate or graduate students in mathematics, physics, or engineering who have a solid foundation in calculus and are interested in furthering their understanding of real and complex analysis.
This textbook covers a wide range of topics in real and complex analysis, including limits, continuity, differentiation, integration, sequences and series, and the theory of complex numbers and functions. It also includes applications to other areas of mathematics, such as Fourier series and differential equations.
While this textbook is primarily designed for use in a classroom setting, it can also be used for self-study. However, it is recommended that readers have a strong background in calculus and a good understanding of mathematical proofs before attempting to study this material on their own.
This textbook provides a comprehensive and rigorous treatment of real and complex analysis, while also including many examples and exercises to aid in understanding the material. It also includes a chapter on the theory of integration, which is often excluded from other textbooks on this subject. Additionally, the author presents the material in a clear and accessible manner, making it suitable for both beginners and more advanced readers.