May have bitten off more than I can chew.

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sotic
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May have bitten off more than I can chew.

Post by sotic »

Pardon me if there's a more appropriate board for this.

Last semester, I was in a complex analysis class where one of the assignments was to give a presentation on some subject in complex analysis. I chose to lecture on Fourier analysis, with an emphasis on music. Steven Krantz attended, said he liked the lecture, and then pitched me a book sale. Interested in learning more, I finally ordered it last week. So now I have this book:

Image

and it seems to be pretty far above my head. If someone could recommend to me how I should approach this book, I would be very grateful.

In the preface, he talks about the history of Fourier analysis, which I can understand up until he mentions singular integrals, Hilbert transforms, Hardy spaces and so on. I've only taken a few calculus classes so I have no idea what any of that means. My hopes are I'll get it by the end of the book. The first actual chapter isn't reassuring so far:
We take it for granted that the reader of this book has some acquaintance with elementary real analysis. ... However, the reader may be less acquainted with measure theory. The review that we shall now provide will give even the complete neophyte an intuitive understanding of the concept of measure, so that the remainder of the book may be appreciated.

[...]

Now let us look at these matters from another point of view. Let Image be any subset. If Image is any open set, then of course we may write

Image

where each Image is an open interval and the Image are pairwise disjoint.
This is where I start to get lost. I get the notion of open sets and subsets, and "pairwise disjoint" makes sense to me, but that big cap operation with all those Image is a foreign thing. The book the goes on to describe m(U) as what I think is supposed to be the sum of the lengths of all the Image. The next few pages are full of operations like Image and it's very intimidating.

My plan right now is to go to the library and start reading textbooks on real analysis and set theory (I haven't had the opportunity to take classes on either of these yet), and perhaps on measure theory if it helps. Is this doable? Is it a lost cause?


(edit: when I was uploading the LaTeX pictures to Tinypic they gave me captchas with Greek letters. Come on.)

Image
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DonRetrasado
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Re: May have bitten off more than I can chew.

Post by DonRetrasado »

http://en.wikipedia.org/wiki/Union_%28s ... ary_unions

You seem to be unfamiliar with the notation. Reading a bit on set theory, or at least having a textbook for reference, might help you read this text. For this kind of analysis, I assume you'd need to at least be able to read basic set notation. The expression you highlighted appears to say something like every open set is the union of all of the intervals it is composed of (of course!). Although I may be totally wrong, as I am bad at set notation and this is the first time I am seeing some of this.
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Peon
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Re: May have bitten off more than I can chew.

Post by Peon »

DonRetrasado has the right idea. Since we're talking about open sets in the real numbers, which are equipped with all sorts of nice things like a metric and a well ordering, there's a lot we can say about the structure of any open set in the reals. Any open set is a (possibly infinite) countable union of disjoint open intervals, and ONLY those. This allows us to throw out a whole bunch of kinds of sets: We can immediately deduce that any sets containing isolated points are not open and that any sets containing half-open/half-closed intervals like (a,b] or [a,b), or containing closed intervals like [a,b], are not open. By the same token, any subset (a,b)∩Q or (a,b)∩I of the rational or irrational numbers is not an open set because it's not an open interval -- there are "holes".

All this stuff seems fairly intuitive in the real numbers. The main reason it's not trivial is because there are many other spaces which can be described with language like "open" and "closed" yet which have wildly differing properties from those of the real numbers.

In the context of your book it just sounds like they're using the notation as a handy characterization of a general open set in the real numbers. The key concept to grasp is that if the number of open intervals that make up the open set is infinite, we can at least say that it's countably infinite. If you've taken some analysis then you might have some idea of how that might become useful in proofs.

This whole open/closed business is what topology is all about. It's a mathematical attempt to isolate the notion of "connectedness" when concepts like distance may not be well-defined.

http://en.wikipedia.org/wiki/Topology
Last edited by Peon on Fri Dec 07, 2012 7:32 am, edited 1 time in total.
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Re: May have bitten off more than I can chew.

Post by Edminster »

thank you for the timely contribution
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Re: May have bitten off more than I can chew.

Post by Peon »

This book looks really cool and I like topology.
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Re: May have bitten off more than I can chew.

Post by Peon »

This branch of math seems pretty cool. After reading around, though, I think I'm gonna check out this book instead of the one in the OP:

http://www.amazon.com/Harmonic-Analysis ... 0691032165

I'm a senior undergrad studying math right now, in the middle of the abstract algebra sequence. Being able to read abstract-ass math is slowly getting easier but I bet this will still read like babbling nonsense at first. Learning how to self-learn is totally a part of being a mathematics student though. We'll see how it goes.
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Re: May have bitten off more than I can chew.

Post by Gilligan »

I felt the same way when I thought that Kant's Critique of Pure Reason would be a good intro piece to Philosophy.
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Re: May have bitten off more than I can chew.

Post by Lethal Interjection »

Gilligan wrote:I felt the same way when I thought that Kant's Critique of Pure Reason would be a good intro piece to Philosophy.
Boy were you wrong. Kant is barely good philosophy.*


*He is good philosophy, I just don't agree with much of it.

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