This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modern physics, and partial differential equations.
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38 of 39 people found the following review helpful:
5.0 out of 5 stars
The essential spectrum of tools for physical observables.,
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This review is from: I: Functional Analysis, Volume 1 (Methods of Modern Mathematical Physics) (vol 1) (Hardcover)
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.
16 of 17 people found the following review helpful:
5.0 out of 5 stars
Excellent,
By A Customer
This review is from: I: Functional Analysis, Volume 1 (Methods of Modern Mathematical Physics) (vol 1) (Hardcover)
We used this book in Math 401 at Princeton. This book is great, well-written, well-problemed. It is curious that the other reviewer complains that there is nothing original in this book, because this subject is older than the author, who is an expert in Schrodinger operators. The parts of the next three volumes which I have read have also been great.
13 of 16 people found the following review helpful:
5.0 out of 5 stars
The best available text on this subject.,
By A Customer
This review is from: I: Functional Analysis, Volume 1 (Methods of Modern Mathematical Physics) (vol 1) (Hardcover)
This superb book may be the best available text on elementary general topology, functional analysis and spectral theory of bounded normal operators on Hilbert spaces. The exercises are excellent as well. Must reading!
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First Sentence:
We assume that the reader is familiar with sets and functions but it is appropriate to standardize our terminology and to introduce here abbreviations that will occur throughout the book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
strong graph limit, norm resolvent sense, strong resolvent sense, diagonalization trick, nuclear theorem, continuous functional calculus, analytic completion, resolvent convergence, abstract measure theory, strict inductive limit, elastic unitarity, dual topology, universal net, bounded linear transformation, weak operator topology, uniform boundedness theorem, spectral theorem, induced unitaries, locally convex space, graph limits, equicontinuous family, mean ergodic theorem, convex spaces, residual spectrum, neighborhood base Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Definition Let, Proof Let, New Jersey, Proof Suppose, Academic Press, Proposition Let, Example Let, Corollary Let, Lemma Let, Van Nostrand-Reinhold, Definition Two, Princeton Univ, Oxford Univ, Rhode Island, Cambridge Univ, Duke Math, Hilbert Schmidt, Nuovo Cimento, Studia Math, Proof By Theorem, Proof See Problem New!
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We assume that the reader is familiar with sets and functions but it is appropriate to standardize our terminology and to introduce here abbreviations that will occur throughout the book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
strong graph limit, norm resolvent sense, strong resolvent sense, diagonalization trick, nuclear theorem, continuous functional calculus, analytic completion, resolvent convergence, abstract measure theory, strict inductive limit, elastic unitarity, dual topology, universal net, bounded linear transformation, weak operator topology, uniform boundedness theorem, spectral theorem, induced unitaries, locally convex space, graph limits, equicontinuous family, mean ergodic theorem, convex spaces, residual spectrum, neighborhood base Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Definition Let, Proof Let, New Jersey, Proof Suppose, Academic Press, Proposition Let, Example Let, Corollary Let, Lemma Let, Van Nostrand-Reinhold, Definition Two, Princeton Univ, Oxford Univ, Rhode Island, Cambridge Univ, Duke Math, Hilbert Schmidt, Nuovo Cimento, Studia Math, Proof By Theorem, Proof See Problem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 53
books:
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- Linear Operators, Part 1: General Theory (Pure and Applied Mathematics, Vol. 7) by Nelson Dunford on 4 pages
- General Topology (Graduate Texts in Mathematics) by John L. Kelley on page 117, page 118, and page 170
- Finite-Dimensional Vector Spaces by P.R. Halmos on page 213, and page 243
- Ergodic Theory (Grundlehren der mathematischen Wissenschaften) by I. P. Kornfeld on page 62, and page 245
- Advanced Calculus, Revised Edition by Loomis on page 31, and page 172
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