John Conway Functional Analysis
Karri Weston
"This book is an excellent text for a first graduate course in functional analysis . . . Many interesting and important applications are included . . . This book is a fine piece of work. It includes an abundance of exercises, and is written in the engaging and lucid style which we have come to expect from the author."
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I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics). What I am looking for is good books that I could understand to go deeper in this areas, what do you recommend? (I can read in Spanish, English, French and German)
$\textbfNote.$ The books which are written in Italics are the ones which I have read partially. The ones which are not in Italics are the ones which I have come to know (by friends, teachers) are good books in Functional Analysis. Also, I really don't know which publisher actually publishes the book in foreign edition written by Kesavan and Bhatia.
Since you read German, my favorite is Funktionalanalysis by Dirk Werner. It's not necessarily comprehensive, but it covers a lot, has extensive historical remarks, and is extremely well-written -- I find it more readable than most math books in English (my first language).
There's no reason to listen to me either, but for delving a bit deeper, you might want to check out T. W. Krner's Fourier Analysis. The book consists of very short (often just a couple of pages) chapters which contain gems like computing the age of the Earth.
Apart from the classics already mentioned (Yosida, Brezis, Rudin), a good book of functional analysis that I think is suitable not only as a reference but also for self-study, is Fabian, Habala et al. Functional Analysis and Infinite-Dimensional Geometry. It has a lot of nice exercises, it's less abstract than the usual book and provides a lot of "concrete" theorems.
I'd recommend the Dunford and Schwartz. It's a classic. It's huge -- three volumes. But you don't have to read the whole series cover-to-cover. If you read half of the first volume, you'll learn about as much as reading many other books on functional analysis. Volume 1 alone is big, but it's easy to read for a book on its subject.
I personally like a recent book of Helemskii Lectures and Exercises on Functional Analysis. One of the differences with other books on the subject is that it uses the categoricalpoint of view. The author starts with a very brief introduction to the category theory and uses this language throughout the book. It's a sort of modern core of FA book, with a sidelines to some physics applications and of historic nature, a terse advertisement of the quantum functional analysis and so on (but there is no measure theory, Radon Nikodym theorem etc. which are elaborated in many excellent old textbooks.) Also it gives somewhat broader picture of FA sketching some directions and stating from time to time theorems without proofs 'that every student should know'.
is quite comphrensive and contains beside standard functional analysis more advanced sections on the theory of locally convex spaces. There is also a German version if you want to improve your German by reading both together.
I have learnt Functional Analysis from Peter Lax himself. His book is his notes. Exactly the same notes as the ones he handed to us. Every chapter consists of one two-hours long lecture in a two semester graduate course on Functional Analysis. There are a few mistakes here and there, but this book is really ALIVE! It is as if you are in a class of Peter Lax! (I should note here that, Lax's book is published a long time after I left Courant, and at that time the recommended textbook was Yosida's book together with Dunford & Schwartz.)
Having said all these, I should add that as an undergraduate student, I had taken two semesters of Functional Analysis which covered a part of Rudin's book. I still use this book sometimes, as some topics are presented in a beautiful way, but I believe that it is far from introductory, as it starts with Topological Vector Spaces, and it takes a while before normed spaces are mentioned.
I second Reed and Simon's methods of mathematical physics. However, if you are interested primarily in the applications of functional analysis to PDE, for the most part a couple of appendices of Evans' book suffice in my opinion.
The best book, I have read it in the past, that I think (I'm not a professor) fits this need is the book in Spanish by Antonio Aizpuru Toms from Universidad de Cdiz with title Apuntes incompletos de anlisis funcional, Editorial UCA (2009).
John Bligh Conway (born September 22, 1939) is an American mathematician. He is currently a professor emeritus at the George Washington University. His specialty is functional analysis, particularly bounded operators on a Hilbert space.
Conway earned his B.S. from Loyola University and Ph.D. from Louisiana State University under the direction of Heron Collins in 1965, with a dissertation on The Strict Topology and Compactness in the Space of Measures.[2] He has had 20 students who obtained doctorates under his supervision, most of them at Indiana University, where he was a close friend of mathematician Max Zorn. He served on the faculty there from 1965 to 1990, when he became head of the mathematics department at the University of Tennessee.
This book covers topics appropriate for a first-year graduate course preparing students for the doctorate degree. The first half of the book presents the core of measure theory, including an introduction to the Fourier transform. This material can easily be covered in a semester. The second half of the book treats basic functional analysis and can also be covered in a semester. After the basics, it discusses linear transformations, duality, the elements of Banach algebras, and C*-algebras. It concludes with a characterization of the unitary equivalence classes of normal operators on a Hilbert space.
The book is self-contained and only relies on a background in functions of a single variable and the elements of metric spaces. Following the author's belief that the best way to learn is to start with the particular and proceed to the more general, it contains numerous examples and exercises.
Functional analysis provides an abstract framework for the study of many questions in analysis. A basic idea behind functional analysis is to regard the objects of interest, for example sequences or functions, as points in a vector space. To obtain a meaningful theory, one endows the vector space with a suitable norm, which makes it possible to talk about concepts such as convergence or continuity. Thus, functional analysis blends together ideas from linear algebra and from analysis.
Dirk Werner - Funktionalanalysis
John B. Conway - A Course in Functional Analysis
Peter D. Lax - Functional Analysis
Gert K. Pedersen - Analysis Now
Walter Rudin - Functional Analysis
Orr Moshe Shalit - A First Course in Functional Analysi
Each of these books contains way more material than we can possibly cover in two months. Central topics from the functional analysis that we will need are the following:
Hilbert spaces, operators on Hilbert spaces, spectral theorem, continuous functional calculus.
Then I learned that standing waves on a drum were related to functional analysis, and then that quantum mechanics (as taught in Dirac's classic book) is nothing but functional analysis, with the structure of the atom and the nature of particles all tied up with functional analysis.
I'd love to see a textbook for functional analysis that proceeds through the theory while giving numerous real-life applications. As of now, Dirac's book on quantum mechanics is the best example I can think of, giving a lot of background theory.
Its main aim is to provide a sound mathematical background for the methods used in quantum mechanics, but it serves well as a textbook on functional analyis (for example, it covers some measure theory, the dual of banach spaces with the Hahn-Banach theorem, locally convex spaces).
The theorems are usually rather brief and might require expansion on the lecture. For instance, I remember a lecture (from volume 4) where we did a proof for 2x90 minutes - the book's version of the proof was only half a page, but iirc, this is a rather extreme example.
Another book worth looking into might be Mathematical Methods in Quantum Mechanics by Gerald Teschl - it's influenced by Reed/Simon as well as the German books of Weidmann, but a bit more recent. On his webpage G. Teschl provides a link to the pdf version of the book, if you want to have a peek at it. Like Reed/Simon, it is a proper functional analysis textbook, aimed at applications in theoretical physics.
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