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Should I take Real Analysis?

I'm surprised no one mentioned the importance of real and funcional analysis in understanding theoretical foundations of probability and stoch calc - IMO some concepts are impossible to grasp without proper analysis background.

BTW I learned Dedekind cuts on my first undergrad year and thought they really sucked, nevertheless I sometimes wish I'd got my BSc in pure math instead of applied.
 
Jacek Podlewski said:
I'm surprised no one mentioned the importance of real and funcional analysis in understanding theoretical foundations of probability and stoch calc - IMO some concepts are impossible to grasp without proper analysis background.
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Functional analysis is the basis for many other fields; for example, numerical analysis (as opposed to numerical methods).

Many structures are subsumed by Hilbert and Banach spaces.

yeah, Dedekind cuts :D hard stuff.
 
Zeuge said:
Many of those concepts are covered in our honors calculus sequence which is a prerequisite to the real analysis sequence. In RA we go over things like metric and topological spaces, and the first class in the sequence ends in the p-adic completion of Q.
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If the basic ideas of real analysis are covered in the calculus sequence, those who've taken the courses should not have any difficulty with the generalisation to metric and topological spaces: after all the definition of continuity in terms of open balls pulling back to open balls (or open sets pulling back to open sets) is just a straightforward generalisation of the epsilon-sigma definition. Likewise, those who've been exposed to Cauchy sequences should be quite comfortable seeing another way of completing Q. The big leap comes from technique-based calculus to proof-oriented courses.
 
bigbadwolf said:
If the basic ideas of real analysis are covered in the calculus sequence, those who've taken the courses should not have any difficulty with the generalisation to metric and topological spaces: after all the definition of continuity in terms of open balls pulling back to open balls (or open sets pulling back to open sets) is just a straightforward generalisation of the epsilon-sigma definition. Likewise, those who've been exposed to Cauchy sequences should be quite comfortable seeing another way of completing Q. The big leap comes from technique-based calculus to proof-oriented courses.
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I think you underestimate how notoriously difficult the sequence can be at some schools, that's why I said depending on where he is, the OP's mileage may vary.
 
Daniel Duffy said:
Functional analysis is the basis for many other fields; for example, numerical analysis (as opposed to numerical methods).
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And is a great piece of beutiful math itself; I happened to write my BSc thesis in operator theory and really enjoyed studying the field (as opposed to Dedekind cuts for example :p) . The main point of my previous post was to show how beneficial good foundations in analysis could be for people intersted in mathematical finance.
 
Jacek Podlewski said:
I happened to write my BSc thesis in operator theory and really enjoyed studying the field.
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There are topics accessible to upper-division undergrads -- for instance, the Spectral Theorem for compact self-adjoint operators on Hilbert space, which is a generalisation of the finite-dimensional Spectral Theorem to the infinite-dimensional Hilbert space setting. The last third of Simmons' "Introduction to Topology and Modern Analysis" covers this (as well as Banach algebras).
 
Banach algebras are awesome, and I ain't saying that just from pure Polish patriotism :) sorry for the OT but I couldn't help expressing my feelings :)
 
bigbadwolf said:
I agree. He who knows nothing of Banach algebras is uncivilised. I sneer at all this talk of the difficulties of elementary analysis. The real math starts at a different level.
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why do you sneer at the talk of difficulties of elementary analysis? analysis is a complete revolution in how most students think about mathematics. Math is just a set of building blocks and it's annoying comments like this(weak attempts at ego boosting) that push people away from the field. Real study of math begins with passion not at a level of education.
 
JesseLevermore said:
why do you sneer at the talk of difficulties of elementary analysis? analysis is a complete revolution in how most students think about mathematics. Math is just a set of building blocks and it's annoying comments like this(weak attempts at ego boosting) that push people away from the field. Real study of math begins with passion not at a level of education.
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My comments here are not serious and I was curious to see it anyone would take the bait. You did. Analysis, incidentally, is not a revolution -- at least not by itself. Linear algebra, abstract algebra, topology, algebraic curves, diff eqs -- all can lay a claim to transforming a math student's thinking. For myself, the single biggest change occurred in linear algebra.
 
Banach rocks. He was a great mathematician.

"Good mathematicians see analogies. Great mathematicians see analogies between analogies."
 
Most of Polish scientists rather agree Banach was the greatest we ever had :) by the way, some of you here might not know that it wasn't him who introduced Banach algebras but they were named that way in his honour.
 
bigbadwolf said:
My comments here are not serious and I was curious to see it anyone would take the bait. You did. Analysis, incidentally, is not a revolution -- at least not by itself. Linear algebra, abstract algebra, topology, algebraic curves, diff eqs -- all can lay a claim to transforming a math student's thinking. For myself, the single biggest change occurred in linear algebra.
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The bait.....right.........Seemed more like another errant comment from someone with no filter from brain to mouth. But we can pretend.

To the OP I hope you have been encouraged to take the course, and do not let people like the above sway you from your pursuits in upper division mathematics. The ones who make the most noise always produce less(I can almost guarantee there will be someone like this in your course); well just imagine if they could stop talking and start doing! Good luck.
 
bigbadwolf said:
My comments here are not serious and I was curious to see it anyone would take the bait. You did. Analysis, incidentally, is not a revolution -- at least not by itself. Linear algebra, abstract algebra, topology, algebraic curves, diff eqs -- all can lay a claim to transforming a math student's thinking. For myself, the single biggest change occurred in linear algebra.
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Well I think it's quite individual. For me analysis was a revolution, similar for most of my peers. Linear Algebra was very straightforward, since there is a large component of computation to get to grips with in the beginning - which is what most students are used to from high-school maths. Further linear algebra can get very abstract, but I did that after I did analysis so the shock wasn't as great.
 
My first eye-opening experience was when I took Abstract Algebra after LA in undergrad. My brain got some real workout after the first few lectures.
I took RA way after that in grad school so the shock factor was long gone but the materials were not easy to say the least.
I enjoy number theory courses the most as their concepts come more natural to me. The US education is such that one can get a BS/MS in Math without doing anything that is really challenging.
 
bigbadwolf said:
Why not Stanislaw Ulam?
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I'd really like to avoid any more OT here, but here comes another interesting question :) I think one possible reason why Banach gets more praise than Ulam in Poland is because he spent most of his life in here, was the brightest star of so-called 'Polish Mathematical School' and so on, whereas major part of Ulam's career took place in the US so he'd been less connected to PL.

As far as RA books are concerned - once you get a grasp of the core ideas and feel like challenging yourself a bit, go and look for problem books by Russian authors - there's high probability of finding some pretty insane stuff in there ;)
 
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