Interested in Funtional Analysis

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Im going to start a major in physics and since now im thinking about my final work for the major. I would like to study on my own some functional analysis, but I dont know where to start. What are the prequisites for studying Functional Analysis? and im also curious about what other advanced subjects are usefull to have some knowledge about in order to make a contribution in functional analysis?

Greetings and thanks in advance.
 

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  • #2
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Functional analysis applies to both linear analysis and non linear analysis.

I would venture to suggest the former is your prerequisite and the latter has plenty of opportunity for new contributions.

go well
 
  • #3
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Functional analysis is a very broad field. So the prerequisites pretty much very with what part of functional analysis you want to study...

I suggest that you start functional analysis with the book by Kreyszig. It has very few prerequisites. Only some calculus and linear algebra is needed. Some familiarity with metric spaces wouldn't hurt, but I don't think it's necessairy...

Of course, if you want to do some high-level functional analysis, then knowledge about topology, real analysis and measure theory will be needed. But I don't think you should meet that very soon...
 
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  • #5
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Thanks, The books of Kreyszig looks very good as an introduction to the subject, I can even understand some of it without having much background.

Im considering to read Apostol or Spivak , then to continue with Loomis, Sternberg 's Advanced Calculus, and then to try Rudin's mathematical analysis.

Thanks for the lectures.
 
  • #6
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Im also interested in a short introduction to set theory. Im thinking about buying Set Theory and Metric Spaces by Kaplansky. What would be other sugestions for a short introduction to set theory?

Thanks
 
  • #7
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I can't comment on Kaplansky. But the best introduction of set theory that I've seen is in the topology book by Munkres. Only the first chapter is about set theory, but it contains all the information about set theory that you'll ever need...
 
  • #8
mathwonk
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Functional analysis usually means infinite dimensional linear algebra. Topology plays a big role, as there are many different ways to put a norm on an infinite dimensional space, and many yield different notions of continuity.

With a topology, one can pass from finite linear combinations to infinite ones, i.e. infinite series, e.g. in "Hilbert space", so calculus is also relevant. Moreover sums may be replaced by integrals.

The analogue of eigenvalues becomes the "spectrum", a subset of the complex plane, and here many books, such as the excellent one by Edgar Lorch, use complex analysis as well.

I am not an expert, and do not know many sources, but i like anything by Sterling K Berberian, as it is always clear as a bell, and also the books by Lorch, and Robert Zimmer.

As stated above one can also consider non linear analysis in infinite dimensions, i.e. calculus on infinite dimensional spaces. E.g. to determine the shortest path between two points you might look at the length function on the infinite dimensional space of paths and try to take its derivative and set it equal to zero. This is also called calculus of variations, and the origin is Euler's differential equation, studied in Courants calculus book vol. 2.

This makes me think that books like Courant Hilbert, methods of mathematicalphysics, may be relevant. Also books like Riesz-Nagy, treat integration theory first and then continue into theor of integral operators.

the name functional analysis presumably stems from the idea of doing analysis on spaces of functions, which are usually infinite dimensional.

It may be advantageous to read modern advanced calculus books like those by Loomis and Sternberg, or Dieudonne;s Foundations of modern analysis, or Lang's Analysis I,II, since those books do calculus in infinite dimensional normed ("Banach") spaces right off the bat.
 
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  • #9
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Thanks, The books of Kreyszig looks very good as an introduction to the subject, I can even understand some of it without having much background.

Im considering to read Apostol or Spivak , then to continue with Loomis, Sternberg 's Advanced Calculus, and then to try Rudin's mathematical analysis.

Thanks for the lectures.
I think that woud take a very long time and is somewhat redundant. In my opinion Loomis and Sternberg is actually at a higher level then Rudin's Principles of Mathematical Analysis (I'm sorry if you meant Rudin's real and Complex). In addition if you read Spivak and Loomis sternberg you will have learned almost everything that covered in Principles. Also none of Rudin's books actually cover much functional Analysis except his book "Functional Analysis."

To learn basic real analysis quickly I'd read Maxwell Rosenschilt's Introduction to Analysis (its only 250 pages and an easy read) Then I'd read Reed and Simon's book on Functional Analysis. It is the first of a four volume series for Physicists but is in fact very popular with Math graduate students. It stresses both applications and proofs and you should be able to read it with a minimum of pure math background.
 
  • #10
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I agree with deluks. Doing Spivak, then Loomis & Sternberg and then Rudin is a bit of overkill. And then you still haven't done any functional analysis!!
 
  • #11
Fredrik
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Kreyszig seems to make it much easier for his readers than most. Conway is ******* ridiculous. Every comment he makes without proof takes hours to get past, sometimes days. It's impossible to read the book unless you know all the basic stuff from topology inside and out. (For example every theorem you can think of that involves compact sets). Now I haven't read Kreyszig, but it seems to me that if you try to read any other book, topology is going to be a much bigger obstacle than any of the posts above are suggesting. The other book I'm reading (Sunder) is much easier, but it doesn't even bother to prove (or even mention that that it can be proved) that a compact Hausdorff space is normal. Apparently you're supposed to figure out all those things on your own.

On the other hand, I don't think you need to know real analysis really well before you study functional analysis. Just have a real analysis book nearby so you can look up stuff about, say, convergence of series.
 
  • #12
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Thanks mathwonk for the explanation I appreciate it .

Yes I agree with Deluks I really have less and less time to study on my own.
A year ago I started reading Apostol but it was complicated, I couldnt manage to solve all the problems specailly the proofs.

Right now im finishing to read my highschool calculus book, a book about proof called" the nuts and bolts of proofs", Im also trying just to solve all the problems in Spivak 's calculus, and Im looking forward to some set theory by Munker's topology. So I think you are right.
I will try to find the books that you mentioned.

I often listen about Rudin's principle of mathematical analysis as being a classic and it is a short book though but I dont know if it is a very complicated one. I used to think that Loomis & Sternberg and then Rudin were totaly diferent subjects though.
 
  • #13
Fredrik
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I often listen about Rudin's principle of mathematical analysis as being a classic and it is a short book though but I dont know if it is a very complicated one.
It is. It's used for a second course in calculus/analysis at the university level, where you go back and cover essentially the same topics as in the first calculus course, but with the emphasis on proofs instead of on how to calculate stuff. I think it takes about twice as much time to get through as many thicker books (about other topics).
 
  • #14
Landau
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Conway is ******* ridiculous. Every comment he makes without proof takes hours to get past, sometimes days. It's impossible to read the book unless you know all the basic stuff from topology inside and out. (For example every theorem you can think of that involves compact sets).
Hm, I actually quite liked Conway! We used it in a second course last semester (basically we covered the entire book, except for the extra 'starred' sections, and the stuff after the Spectral Theorem for normal operators). He does write in a compact style (pun intended), but he also says that he assumes a 'thorough knowledge of measure and integration theory and some point-set topology'. So I wouldn't use it as a first introduction, rather as a second textbook.

I very warmly recommend Pedersen's Analysis Now as a supplement to Conway. He covers all necessary topology in the first chapter, and has a really nice exposition of functional analysis. I think you'll like it, Fredrik. His writing style is great. Conway is more thorough, though.
Now I haven't read Kreyszig, but it seems to me that if you try to read any other book, topology is going to be a much bigger obstacle than any of the posts above are suggesting. The other book I'm reading (Sunder) is much easier, but it doesn't even bother to prove (or even mention that that it can be proved) that a compact Hausdorff space is normal. Apparently you're supposed to figure out all those things on your own.
Rynne & Youngson's https://www.amazon.com/dp/1852332573/?tag=pfamazon01-20 is at a comparable level to Kreyzig. Relatively elementary, but crystal clear.

On the other hand, I don't think you need to know real analysis really well before you study functional analysis. Just have a real analysis book nearby so you can look up stuff about, say, convergence of series.[/QUOTE]
 
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  • #15
Kreyszig is one of the most readable math books out there, you can't go wrong with it. You'll definitely want to pick it up either way if you are continuing in any field where you may need functional analysis.
 
  • #16
Why not do Shilov's course - https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20? You could supplement both the real & functional analysis books with the corresponding texts by Kolmogorov & Fomin. If you're still worried about proofs I'd suggest "Analysis with an Introduction to Proof" along with "Discrete Mathematics Demystified".
Also, https://www.amazon.com/dp/0470107960/?tag=pfamazon01-20 would be useful, there's an intro to topology, Hilbert spaces etc... and it's an elementary book.
 
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  • #17
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I have read some of Discrete mathematics demystified before but I would rather try books that are more technical with proofs.
I was thinking about reading Apostol vol 2 in order to learn linear algebra and some analysis but it is a long reading. I guess it is more convenient to read a book just about linear algebra and one about analysis instead.
Mathematical Analysis: A Concise Introduction, looks pretty good it has a lot of subjects though.
Thanks.
 
  • #18
I liked Discrete Math Demystified because it gives direct elementary proofs illustrating proof
techniques and constructs number systems. If you want something harder you should try
Lay's Analysis book or the Schroder book.

I found a beautiful combination of (free) resources to learn linear algebra that has both
plenty of proofs and enough numerical examples to illustrate the points:

Dawkin's Linear Algebra
http://nptel.iitm.ac.in/video.php?courseId=1097 [Broken]
http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/index.html [Broken]

There is an almost perfect correlation between these three sources, where the videos are
unclear in a certain proof the Dawkin's notes clarify and vice versa, no joke. Same with
the examples. If you started at module 2 in the video courses and did from video 15 to 32
while also doing the corresponding sections in the Dawkins notes & nptel notes you would
have the contents of an elementary linear algebra course done. You could then go onto
Sharipov's & mathwonk's notes & Axler or Hoffman/Kunze. (Note: you could read these
advanced books now obviously, what I gave you above assumes you need a few more
examples than those books give).

Also, this functional analysis book might benefit someone since it looks utterly amazing
& apparently has modest prerequisites.
 
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  • #19
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simmons book topology and modern analysis is excellent for set theory as well as functional analysis
 

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