Question about Real Analysis & Functional Analysis

[ILOVETORONTO]

I am an undergraduate student, and I decide to learn real analysis and functional analysis this term.
And I wanna apply for MFE in the future. However, I have no idea about the application of these two subjects in the field of Financial Engineer. I do hope that I could find the answer in QuantNet !!!
THX!!!

 

[Daniel Duffy]

IMO
In general, you won't need Functional Analysis directly but it is well worth learning. Go for it.
Real Analysis is essential I would say.

Even more important is Numerical Analysis.

 

[ILOVETORONTO]

IMO
In general, you won't need Functional Analysis directly but it is well worth learning. Go for it.
Real Analysis is essential I would say.

Even more important is Numerical Analysis.

THANK YOU VERY MUCH! I'll try to read some books about Numerical Analysis. By the way, could you please recommend some textbooks for me? some elementary.

 

[Polter]

Some subjective recommendations:
http://math.stackexchange.com/questions/50444/teaching-introductory-real-analysis
http://www.ocf.berkeley.edu/~abhishek/chicmath.htm#i:real-analysis
http://www.ocf.berkeley.edu/~abhishek/chicmath.htm#a:functional-analysis

 

[D_chuqu]

Don't worry! In fact, I don't think Real Analysis will be very essential since most of the functions in engineering are Riemann Integrable. But the greatest influence of Real Analysis on MFE would be that the Lebesgue Measure Theory may give you deeper understanding on the probability space, which I think will be very important in dealing with financial problems.

 

[Daniel Duffy]

Don't worry! In fact, I don't think Real Analysis will be very essential since most of the functions in engineering are Riemann Integrable. But the greatest influence of Real Analysis on MFE would be that the Lebesgue Measure Theory may give you deeper understanding on the probability space, which I think will be very important in dealing with financial problems.

That's a good question? How useful is measure theory? How many traders/quants 'use' it?

 

[Daniel Duffy]

http://www.amazon.com/Introductory-Analysis-Dover-Books-Mathematics/dp/0486612260

 

[JesseLevermore]

I can tell you now that real analysis is very, very important if you want to learn Stochastic calculus at the research level. Now if you just take what universities are calling "advanced calculus", then that will not be enough but the concepts of analysis are all over probability theory and stochastic calculus. (I mean logically how can you understand the Lbesgue without Riemann Stieltjes and how can you understand Ito integral without Lebesgue).

You need mainly sigma algebras, this is analysis. You need Borel measure and you need to understand inverse functioin; this is also analysis. You need to understand all the little tricks of analysis too (i.e. the union of open sets is open, finite intersection only of open is open; continuity of functions and the properties). Many probability theory books will gloss over the analysis because it's usually expected but yes you need a lot of it.

 

[D_chuqu]

You are probably right! But I have worked in this field for more than two years. I don't think it is of no use.

 

[bigbadwolf]

I can tell you now that real analysis is very, very important if you want to learn Stochastic calculus at the research level. Now if you just take what universities are calling "advanced calculus", then that will not be enough but the concepts of analysis are all over probability theory and stochastic calculus. (I mean logically how can you understand the Lebesgue without Riemann Stieltjes and how can you understand Ito integral without Lebesgue).

But how to go about learning it? I looked at the links Polter gave and read the first two (on the teaching of real analysis). I know virtually every book listed there. What's being discussed is not what to teach -- on which there seems to be a consensus -- but rather the most effective way of teaching the material (~ the contents of baby Rudin). It seems to me a more fruitful question to ask would be what to teach -- and then the method of teaching would sort itself out. The modern real analysis course is all structure and few applications -- the nature of the reals, the Cauchy criterion, Bolzano-Weierstrass, intermediate value theorem, the Riemann (-Sieltjes) integral, and so on. Applications are sparse. The problem with teaching it is the barrage of decontextualised definitions, lemmas, theorems, and corollaries, for which motivation seems to be absent. Much of the 18th and 19th century math that motivated the development of real analysis is either not taught or (worse) has been forgotten. Yet arguably this is what brings the subject to life.

 

[muttiah]

But how to go about learning it? I looked at the links Polter gave and read the first two (on the teaching of real analysis). I know virtually every book listed there. What's being discussed is not what to teach -- on which there seems to be a consensus -- but rather the most effective way of teaching the material (~ the contents of baby Rudin). It seems to me a more fruitful question to ask would be what to teach -- and then the method of teaching would sort itself out. The modern real analysis course is all structure and few applications -- the nature of the reals, the Cauchy criterion, Bolzano-Weierstrass, intermediate value theorem, the Riemann (-Sieltjes) integral, and so on. Applications are sparse. The problem with teaching it is the barrage of decontextualised definitions, lemmas, theorems, and corollaries, for which motivation seems to be absent. Much of the 18th and 19th century math that motivated the development of real analysis is either not taught or (worse) has been forgotten. Yet arguably this is what brings the subject to life.

I would agree. Real analysis is a beautiful subject but it usually presented without motivation. Although it's a level below Rudin, Spivak's calculus books brings the subject more to live and is an excellent introduction to rigorous analysis.

 

[Polter]

How about Bressoud's books -- in particular, "A Radical Approach to Real Analysis" and "A Radical Approach to Lebesgue's Theory of Integration"? He seems to take the historically-motivated approach you mention:
http://www.macalester.edu/~bressoud/books.html

 

[JesseLevermore]

Illegitimate as it may seem I think the main criterion for a good analysis text is pictures. The concepts of limit points, bounded sequences and countless others come through so clearly in two dimensions that you can almost immediately think of what the proof involves.

This by no means implies that an image based intuition is always correct, but if you can learn through examples with pictures I think you'll have the quickest and smoothest learning experience in analysis possible. I like the Pugh book but there are probably others that accomplish what he does. I know Rudin can be a really difficult read first time through (I've never really spent much time with it though).

It really does seem like a subject in which a good professor can be of so much benefit. The examples, the intuition, the requisite amount of abstraction for the level, it is a subject that does not seem fit for self study. I never really slogged through an entire text, rather read pieces from wherever I might find what I needed for what was asked of me. I think it has allowed me to learn that much quicker since I don't doubt that I could read any text all the way through but what use would I have for the many side topics that a book can cover? And clearly it isn't the role of a beginner to decide what is or is not relevant to a subject matter 300 years in the making.

 

[bigbadwolf]

How about Bressoud's books -- in particular, "A Radical Approach to Real Analysis" and "A Radical Approach to Lebesgue's Theory of Integration"? He seems to take the historically-motivated approach you mention:
http://www.macalester.edu/~bressoud/books.html

Got 'em both. Good books. I also like Peter Duren's Invitation to Classical Analysis.

 

[Daniel Duffy]

But how to go about learning it? I looked at the links Polter gave and read the first two (on the teaching of real analysis). I know virtually every book listed there. What's being discussed is not what to teach -- on which there seems to be a consensus -- but rather the most effective way of teaching the material (~ the contents of baby Rudin). It seems to me a more fruitful question to ask would be what to teach -- and then the method of teaching would sort itself out. The modern real analysis course is all structure and few applications -- the nature of the reals, the Cauchy criterion, Bolzano-Weierstrass, intermediate value theorem, the Riemann (-Sieltjes) integral, and so on. Applications are sparse. The problem with teaching it is the barrage of decontextualised definitions, lemmas, theorems, and corollaries, for which motivation seems to be absent. Much of the 18th and 19th century math that motivated the development of real analysis is either not taught or (worse) has been forgotten. Yet arguably this is what brings the subject to life.

And not forgetting 18th century France (Lagrange, Laplace, Fourier etc.)

Maybe mention Dedekind: he replaced visual handwaving infinitesimal calculus by arithmetic (Dedekind cuts).