Functional Analysis (Methods of Modern Mathematical Physics (Volume 1))

really great book! easy to understand and interesting to read.

Books on mathematical methods "for physicists" are often criticized by their superficiality, a sacrifice deemed necessary for achieving completeness. This one is a glaring exception: the first of a set of 4 (!) volumes dealing with the finest tools for dealing with the delicate mathematical questions in quantum theory - namely, functional analysis. Of course, this sounds rather vague, since quantum physics makes use of functional-analytic tools as diverse as distributions, Hilbert, Banach and locally convex spaces, spectral theory, semigroup theory, operator algebras, etc.However, do not expect ready-brew formulae and cookbook recipes: this book gets his job done at least as well as Rudin, Yosida and Riesz-Sz.Nagy, just to mention the classics. Most theorems are rigorously proved, and although the book becomes more and more biased towards mathematical physics (i.e., methods for proving self-adjointness, analysis of spectra and scattering theory, as stated in the section "Three Mathematical Problems in Quantum Mechanics". These methods occupy most of the three remaining volumes) as it proceeds - this bias becomes the true reason of being for the last two volumes - this particular volume has precisely the most useful stuff: metric, Banach, topological, locally convex, and Hilbert spaces, bounded and unbounded operators. A supplement extracted from the second volume with the basics of Fourier transforms makes it self-contained as a monograph.However, the best things, that make this book nearly unbeatable, are the several wisely chosen examples and counterexamples, the notes at the end of each chapter and the wonderful - and useful - exercises. Many working mathematicians I know use this book seriously in their research and their courses in Functional Analysis - a fact that cannot be underestimated and will hardly be equaled by any book on mathematical physics.If you work on (axiomatic) quantum field theory you may also want to keep an eye on the second volume of the set, "Fourier Analysis, Self-adjointness", which is a bit more specialized but just as wonderful.

This is the best functional analysis book for beginners, in my opinion. It is written for people that are interested in functional analysis as a tool for differential equations. What makes it different from other books on this subject are the numerous examples and applications to differential equations. Highly recommended.

This is an excellent text on functional analysis. I read a few books through the library when learning the subject and really loved the clarity of Reed-Simon the most. Also, the exercises are great.However, the actual book that I received felt almost like photocopy quality and was difficult to read. The whole point of purchasing such an expensive text is to have it in your hands without the strain of staring at a computer screen. Elsevier did a downright crappy job with the new version (not the old maroon one).

Functional analysis can be difficult. But, does it have to be this difficult ? Prerequisites: “sophistication acquired in a typical undergraduate mathematics education in the United States.” (introduction, 1972 edition). I do not claim to know the “typical” undergraduate mathematics major (even less so, if a student of Princeton University 1972), but, I would be surprised if the “sophistication” of a "typical" undergraduate student met the minimum standard required to understand this text. I am hardly a stranger to the “definition-theorem-proof” format (which this book takes), yet, the denseness of this book is forbidding (contrast: Naylor and Sell, who utilize that format, it being the ideal introduction to functional analysis: Linear Operator Theory in Science and Engineering).(1) Initial chapter (30-page review of preliminaries) is comparable to chapter one of Lieb and Loss (Analysis). Hilbert Space (chapter two) is easier to follow than the “preliminaries,” concluding with an interesting section on ergodic theory (further developed in chapter six). Problem #14 entails use of Cesaro summability. Happily, chapter three, Banach Space, really a continuation of chapter two, is easier after all that.(2) The book gets less difficult as you progress through the chapters. Initial sections of each chapter are "basics,” later sections are “specialized.“ Here is one exercise (topological vector spaces): “prove the intermediate-value theorem of freshman calculus.” (chapter four, problem #3, page 120, this is the last part of a six-part exercise). Exercises conclude each chapter. The exercises consist mostly of devising mathematical proofs.(3) Meet convexity (chapter four, page 109), then get a full-fledged elaboration (chapter five, locally convex spaces). I highlight chapter five, as it is a favorite: An interesting discussion of bootstrap equations, then, rather brief, presentation of scattering (pages 155-161).(4) Informative “Notes” concludes each chapter. Notes are ancillary to the primary text and serve as a guidepost to research and open questions. For example: Birkhoff and von Neumann's “The Logic of Quantum Mechanics (1936)” is referenced (page 311). Roberts' “Rigged Hilbert Space in Quantum Mechanics (1966)” is referenced (page 244). Read those two papers online. If you desire an introductory paper, Madrid: "Role of Rigged Hilbert Space in Quantum Mechanics," (2005, arXiv).(5) Lieb and Loss write that “most texts make a big distinction between real analysis and functional analysis, but we regard this distinction as mostly artificial.” (1997, preface, Analysis). Lieb and Loss emphasize inequalities (see their proofs: pages 38-41). It is of interest to compare Reed and Simon (page 68, Minkowski and Holder inequalities) to Lieb and Loss. Choose the approach that works best for you.(6) Comparison is in order for this statement: “We have already intimated that measures on ‘R’ are associated with quantum mechanical Hamiltonians, and they, in turn, arise from certain positive linear functions and the use of the Riesz-Markov theorem.” (page 107). Compare that statement with words found in Lieb and Loss (page 151).(7) A book, “Introduction to Axiomatic Quantum Field Theory “ (1975, Bogolubov, Logunov, Todorov), describesthe basics of functional analysis, distribution theory and Fourier transformations in its first 100 pages. My advice: study those hundred pages as a preliminary to Reed and Simon. I can think of no better preliminary.(8) The crowning achievement here is chapter eight, unbounded linear operators. It is a good one. Of course, the previous seven chapters are prerequisite to its understanding. Read: “formal calculations can be misleading.” (page 270). Naylor and Sell also conclude their fine introductory text with that topic.(9) I conclude my review. If you delight in a “definition-theorem-proof format,” then this is your book. Examples abound, exercises too (most are asking for proofs). The book works best as a reference, but as a first exposure the book will prove wanting. My opinion: it is lacking in pedagogic strategy. The text by Lieb and Loss, Analysis, is at the same level of sophistication with a different selection of topics (I like it quite a bit). Pick and choose !

This books is an exposition to hilbert space theory. It contains a demonstration of the spectral theorem.It is a deceiving book: while clearly written, there is nothing original in it.

Dieses Buch ist ein absolutes Standardwerk und eines der besten Bücher zur Funktionalanalysis! Ich kenne nur ein einziges anderes Buch, das die Funktionalanalysis in vergleichbarer Breite behandelt: Den Werner. Diese sind die einzigen mir bekannten Bücher, in denen lokalkonvexe Räume und unbeschränkte Operatoren behandelt werden. Ich betrachte die beiden Bücher als gleich gut: Der Werner ist oft etwas gründlicher (an Stellen, wo Beweise im Reed/Simon verkürzt dargestellt sind) und behandelt auch z.B. Sobolevräume, die im Reed/Simon leider fehlen. Dafür erfährt man im Reed/Simon viel über Anwendungen in der Physik (man darf nicht vergessen, daß es sich um den ersten Band einer Lehrbuchreihe handelt!) und oft ist die Darstellung im Reed/Simon etwas übersichtlicher.Die Bewertung bezieht sich auf den Inhalt. Was die Druckqualität angeht, kann ich den anderen Rezensionen nur beipflichten. Jedoch muß ich sagen: Seitenzahlen sind oft schlecht lesbar und stellenweise unkenntlich. Dennoch ist die Qualität ausreichend, um gut mit dem Buch arbeiten zu können. Formeln sind kein Problem!Der Preis ist stolz und ich würde dem Kaufinteressenten raten, auch mal einen Blick auf den neuen Band Operator Theory von Barry Simons Analysis-Kurs zu werfen. Beim Inhalt scheint es große Überschneidungen zu geben und der neue Band ist etwas günstiger (obwohl immer noch teuer) und zudem in LaTeX gesetzt. Ich besitze ihn allerdings nicht, da es in noch nicht gab, als ich meinen Reed/Simon gekauft habe.

This book is fantastic. It is beautifully written. The authors motivate all of the theory, making it seem as natural as possible. This aspect seems to be missing from all the other standard Functional Analysis texts I have read. I feel the authors really want to share the ideas with me. All too often authors and teachers seem to prevent you from understanding, perhaps because they are not willing to do the hard work of explaining clearly, or perhaps because they want to make themselves seem more intelligent than you. In fact, Reed's and Simon's generous, painstaking, welcoming approach means that they are brighter by far than their competitors ! I borrowed this book from the library a few days ago, since when I haven't put it down ! I knew straight away that I needed to buy my own copy.

Works fine.

Der erste Band der Monographie Serie „Methods of Modern Mathematical Physics“ von Michael Reed und Barry Simon zur Funktionalanalysis erschien 1972 in der ersten Auflage, 1980 wurde eine überarbeitet und erweiterte Edition herausgegeben; seitdem hat dieses Buch zahllosen Vorlesung für Physiker und Mathematiker als Grundlage gedient und Generationen von Studenten haben dieses Werk schätzen gelernt.Einer der Gründe für diesen Erfolg ist wohl die gelungene Stoffauswahl in Verbindung mit der eleganten, kristallklaren Art der Darstellung; das Buch deckt neben den Standard Themen der Funktionalanalysis der Mathematischen Physik, wie Banach und Hilberträume, Operatoren und deren Spektraltheorie, auch Themen, wie allgemeine Topologie und topologische Maßtheorie sowie Lokal konvexe Räume, ab.Dabei bleibt die Darstellung konzentriert, unnötiger Formalismus und überladene Bezeichnungen werden vermieden, der Leser wird stets jeweils auf möglichst direkt Wege zu den zentralen, tiefen Theoremen des jeweiligen Gegenstands hingeführt, die Beweise sind knapp aber präzise. Die behandelten Themen sind – bis auf wenige Ausnahme, wie etwa der Konstruktion von Haar Maßen – sind in sich abgeschlossen; das macht das Buch auch als Referenz- und Nachschlagewerk wertvoll. Die vorliegende erweiterte Ausgabe hat dabei noch einige Lücken, durch eine Anzahl von Supplements, geschlossen.Jedes Kapitel wird mit einer Sammlung 'Notes' mit Referenzen und historischen Anmerkungen, und einer Reihe von Aufgaben beschlossen, die Aufgaben beinhalten nicht nur Beispiele, sonder oft auch weiterführendes Material, das im Haupthandlungsstrang ausgespart wurde.In der summa: ein überaus durchdachtes und gut lesbares Standardwerk.

Das Buch von Reed/ Simon ist vom Inhalt her erstklassig und absolut zu empfehlen für das Verständnis der mathematischen Grundlagen der Quantenmechanik.Die angeblich überarbeitete Neuauflage lässt jedoch mehr als zu wünschen übrig. Die Seiten wurden nicht neu gedruckt, sondern scheinbar nur kopiert oder eingescannt und ausgedruckt. Viele grafische Darstellungen sind dadurch beeinträchtigt, ebenfalls sind viele Indices sogar unlesbar. Hinzu kommt, dass man sogar beim Druck des Schutzeinbandes nicht in der Lage war "revised" richtig zu schreiben. Auf dem Buchrücken fehlt da ein "i".Fazit: Für den Preis ist das trotz des Inhaltes nur ein mangelhaft, da erwarte ich vom Verlag doch mehr Professionalität.