Which is the best book of functional analysis? How to learn functional analysis efficiently?

QuoteOriginally posted by: Vincent Which is the best book of functional analysis? How to learn functional analysis efficiently?I'm not an expert, but maybe you wish to take a look at a recent book (2002) by Peter D. Lax. He's one of the greatest applied mathematicians of the 20th century, and (if I remember correctly) it received some very good reviews on amazon.PS If you are at or close to a math dept, the best book may be the one used by the functional analysis lecturer, so you can ask him questions. But be careful, because it's a subject that attracts a large number of brain-dead third-rate mathematicians, who get very high on fomalism and jargon. That's why I recommended Lax's book. He's too smart and too broad to get stuck in jargon.

I went to my shelf, and the book that I would recommend, and it happens to be re-printed by dover, is D.H. Griffel's Applied Functional Analysis. Here is the descriptioon from amazonA stimulating introductory text, this volume examines many important applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Detailed enough to impart a thorough understanding, the text is also sufficiently straightforward for those unfamiliar with abstract analysis. Its four-part treatment begins with distribution theory and discussions of Green's functions. Essentially independent of the preceding material, the second and third parts deal with Banach spaces, Hilbert space, spectral theory, and variational techniques.The final part outlines the ideas behind Frechet calculus, stability and bifurcation theory, and Sobolev spaces. 1985 ed. 25 Figures. 9 Appendices. Supplementary Problems. Indexes.

First of all I would highly recommend a book by Kolmogorov, Fomin “Elements of the Theory ofFunctions and Functional Analysis”. It’s a classic and a few generations of (post)-Soviet mathematicians (and probably not only post-Soviet) have been brought up on this book. The main prerequisite is, of course, a good knowledge of analysis and linear algebra (I mean, with prior knowledge of finite-dimensional operators it would be easier to jump into infinite-dimensional ones).Secondly, I would also advise a book by Berezansky, Sheftel, Us “Functional Analysis”, Vol. 1-2. It was used in my course last semester at Kiev University (Ukraine) and actually Ukrainian professors wrote it. It lacks some advantages of the previous book but it’s more modern, there is more material in it and there are lots of good and not-so-hard exercises. I don’t know much about a translation, but the original is well written. There is a contents on NY’s Springer page.I personally think that the best way to learn FA is to do a lot of exercises, starting from simple and moving to harder. But of course, as I mentioned before, one has to know a lot before starting to learn this advanced subject.Good luck!

QuoteOriginally posted by: DebutI would highly recommend a book by Kolmogorov, Fomin “Elements of the Theory of Functions and Functional Analysis”. It’s a classic and a few generations of (post)-Soviet mathematicians (and probably not only post-Soviet) have been brought up on this book. The main prerequisite is, of course, a good knowledge of analysis and linear algebra (I mean, with prior knowledge of finite-dimensional operators it would be easier to jump into infinite-dimensional ones). Thanks for reminding me of that. I have it, but I haven't read it.

HiI'd recommend Introductory Functional Analysis with Applications (Wiley Classics Library) by Erwin Kreysig. This book served me well through my final year of undergraduate studies and during my graduate studies in Applied Maths (specifically infinite dimensional linear operators). The book is very easy to get into with a first year undergraduate grounding in group theory and calculus.Hope the following link works:http://www.amazon.co.uk/exec/obidos/ASI ... 5213425Let me know what you think - good luck!

I believe I have a copy of JB Conway's 'Func Anal' which I will offer at a very decent price point net of shipping

I would suggest the classicalM. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, ISBN: 0-12-585050-6M. Reed and B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis , Self-Adjointness, Academic Press, ISBN: 0-12-585002-6and W. Rudin, Real and Complex Analysis, McGraw-Hill College, ISBN: 0070542341W. Rudin, Functional Analysis, McGraw-Hill Science/Engineering/Math, ISBN: 0070542368(the two from Rudin are (o were.. long ago!) also available in paperback from Tata McGraw-Hill Publishing)Have fun!Marco

Here's another question:Where do people need the ideas, theorems and results of Functional Analysis in their daily work in Financial Engineering? Of course, it's great knowing FA but is is directly relevant? I am not talking about measure theory, Fourier analysis and so on but am referring to subjects like:Hilbert and Banach SpacesHahn-Banach theoremLP and Sobolev spacesApplied Functional Analysis (e.g. Finite Element Method)You need all this stuff for PDE but I have not seen it being used in other areas of FE, really.Any views?Most practical applications in FE tend to work in finite-dimensional spaces, so FA is not 100% necessary?

QuoteOriginally posted by: LongThetaQuoteOriginally posted by: Vincent I'm not an expert, but maybe you wish to take a look at a recent book (2002) by Peter D. Lax. He's one of the greatest applied mathematicians of the 20th century, and (if I remember correctly) it received some very good reviews on amazon..And a brilliant numerical analyst to boot. Remember the Lax Equivalence Theorem that states: given a consistent finite difference scheme to a initial value problem, stability is necessary and sufficient for convergence.

- Functional Analysis by Peter Lax- Applied Functional Analysis by D. H. Griffel- Elements of the Theory of Functions and Functional Analysis by Kolmogorov and Fomin- Introductory Functional Analysis with Applications by Kreyszig- A Course in Functional Analysis by Reed and Simon

QuoteWhere do people need the ideas, theorems and results of Functional Analysis in their daily work in Financial Engineering? Of course, it's great knowing FA but is is directly relevant? I am not talking about measure theory, Fourier analysis and so on but am referring to subjects like:I'm reading Rudin and Lax at the moment....no idea if it will ever be useful in daily work, but it is very interesting. btw... are there any solutions out there for the excercises in these books.Some of them are quite tough (for my humble brain)

QuoteOriginally posted by: CuchulainnHere's another question:Where do people need the ideas, theorems and results of Functional Analysis in their daily work in Financial Engineering? Of course, it's great knowing FA but is is directly relevant? I am not talking about measure theory, Fourier analysis and so on but am referring to subjects like:Hilbert and Banach SpacesHahn-Banach theoremLP and Sobolev spacesApplied Functional Analysis (e.g. Finite Element Method)You need all this stuff for PDE but I have not seen it being used in other areas of FE, really.Any views?Most practical applications in FE tend to work in finite-dimensional spaces, so FA is not 100% necessary?the standard proof of the fundamental thm of asset pricing uses the Hahn-Banach theorem

QuoteOriginally posted by: CuchulainnHere's another question:Where do people need the ideas, theorems and results of Functional Analysis in their daily work in Financial Engineering? Any views?Most practical applications in FE tend to work in finite-dimensional spaces, so FA is not 100% necessary?Another route to FA in Finance: 99% of the 'models' in finance are Markov processes that live on some continuous, unboundedstate space -- typically products of R+ = (0, \infty). The processes have transition functions.The transition functions satisfy forward and backwards equation.Then, the question becomes: what are the natural 'function spaces' for the solutions to these fundamental evolution equations? regards,

I agree with Alan. Brownian motion and its variants (with mean reversion, multi-factor, etc) used in most of finance generate diffusion equations. The solution of a diffusion equation with a point source (Green function or transition function) is a Feynman/Wiener path integral, actually a functional. The construction is straightforward. Stochastic equations containing Wiener measures provide the dynamics. You start with the probability integrals for the Wiener measures at discretized times, and then insert the stochastic equation at each time as a Dirac delta-function constraint. Calculations are done in discretized time, and then the continuum limit is taken for the results. You can also take the continuum limit formally to get an infinite iterated functional integral. You prove by direct calculation that the path integral satisfies the appropriate diffusion equation associated with the stochastic equation. Analytic solutions, when they exist, can be derived by evaluating the path integral exactly, usually just by completing the square and doing standard calculus integrations. Numerical methods can be regarded as approximations for evaluating the path integral. Thus the functional path integral provides a straightforward and unifying methodology in finance. See my book for details, Chapters 41 to 45. The formal mathematical properties of path integrals in finance are simpler than for quantum mechanics, since there are no oscillations here (i.e. finance is "Wick-rotated"). Those interested in a formal treatment might look at for example the book "Quantum Physics - A Functional Integral Point of View" by Glimm and Jaffe. -----------