Elements of the Theory of Functions and Functional Analysis (Dover Books on Mathematics) by A. N. Kolmogorov (PDF)

Description

Originally published in two volumes, this advanced-level text is based on courses and lectures given by the authors at Moscow State University and the University of Moscow.Reprinted here in one volume, the first part is devoted to metric and normal spaces. Beginning with a brief introduction to set theory and mappings, the authors offer a clear presentation of the theory of metric and complete metric spaces. The principle of contraction mappings and its applications to the proof of existence theorems in the theory of differential and integral equations receives detailed analysis, as do continuous curves in metric spaces — a topic seldom discussed in textbooks.Part One also includes discussions of other subjects, such as elements of the theory of normed linear spaces, weak sequential convergence of elements and linear functionals, adjoint operators, and linear operator equations. Part Two focuses on an exposition of measure theory, the Lebesque interval and Hilbert Space. Both parts feature numerous exercises at the end of each section and include helpful lists of symbols, definitions, and theorems.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a combination of two books: functional analysis and real analysis. The price is affordable but the print quality is just OK. Some terminologies are kind of “old-fashioned”. There are also typos (e.g. < should be <= on page 77 for the definition of bounded). The writing styles are intuitive and enlightening. ⭐I've gotten through most of the first volume (First volume is on linear spaces, second one on measure theory, both volumes are included in this book) and this has been an excellent reference and supplement to the text we are using. The terminology is somewhat outdated: for example there is mention of contact points (which would just be called a limit point today) and the codimension of a subspace is called its deficiency (which some people still use actually), but again as a reference and supplement to a more recent text it is superb. In particular I found its treatment of the contraction principle to solving nonlinear differential equations very useful in understanding that found in Royden. Eventhough Royden is the text used in my graduate class, its obvious the professor is using this book (probably a first edition of it lol) because the order in which he presents the material is pretty much the same as presented in this. All in all it is far too dense to be wielded properly by a beginner in functional analysis and operator theory but for the price it is a must to have even if just to get a different point of view on something you might have trouble understanding somewhere else. I cannot comment on the second volume as I haven't gone through it at all. ⭐This is another wonderful Dover classic; so a reprint of a first edition that later became a classic.The level is upper level undergraduate and beginning graduate. But it is also a great book for someone who is looking for a quick overview of basic tools in function theory, and functional analysis.Grant you that a lot has changed since the first printing in the 1940's.Key topics in the book are as relevant now as then, e.g., (sample from the contents) Normed and Topological Linear Spaces; and their duals; Linear Operators; Differential Calculus and integrals; Measurable Functions, Integration theory; Summable functions; spectral theory; Trigonometric series, Fourier transformation; Fredholm Integral Equations. Andrei Kolmogorov was a true pioneer in what is now modern mathematics, and its neighboring areas: probability theory, stochastic processes, harmonic analysis, information theory, dynamical systems.One point where the coverage is limited is in its focus on linear theory. With hindsight we now know that non-linear features must be used in models for turbulence and other areas of applications. The book is great, nonetheless, for selfstudy, or for a supplement to anyone of a number of courses in mathematics. Review by Palle Jorgensen, December 2011. ⭐Esta edición en inglés no está completa, le faltan muchos capítulos, esta edición de Dover es de 288 páginas tengo una copia de este libro en español de la editorial MIR y tiene 536 páginas.No recomiendo esta edición ya que está muy resumida y le falta mucha información en comparación a la edición en español.The main reason I'm writing this review is that I think it would be useful to know that the two volumes bound as one that you are getting have their pages numbered separately. The first one is 129 pages and the second is 128, which is the page count that Amazon lists. I was concerned about whether I was getting all the Kolmogorov I was entitled to since other editions listed over 200 pages, but it's all there (well, another reviewer said that the most recent Russian edition is more expansive... you're getting everything that's been translated). I don't have much else to add; the book is excellent and, although some background in analysis would help, the writing is so clear that I think you could follow it without any preparation (this is coming from a physics major who hasn't taken any pure math beyond 300 level vector analysis). ⭐I got my math degree in Ukraine and I read another books of Kolmogorov and Fomin. This is first their work in English for me. And both parts were impressed me. I got this book because I did not understand why price is so small. These mathematicians are both great and the book can be sold in similar price as rest of math books of same category.Bottom line. Book is great and material well balanced. Practice problems were prepared to improve understanding of covered material.As result this book is always with me. I do not think that I can add more to that. ⭐Only thing wrong about it is it could have been organized better. It combines the two volumes of the original work but the volumes are distinct within the book, each with its own page numbers and indices, which is weird to say the least. They could have easily redone just the page numbers and combined the indices and TOC and let everything else be and it'd have been even better. Still, I cant give it anything less than five stars because the book is so good and it's Dover so the price is great. ⭐I can't use this Kiindle edition. The pages seem to be in completely random order. They are also formatted sideways. I will return the book if I can figure out how to do that. ⭐An excellent introduction to the basic principles of Functional Analysis: elements of the theory of sets and functions, metric spaces, normed spaces, introduction to the theory of distributions and spectral theory; in the second part of the volume, measure theory and abstract Lebesgue integration, Hilbert spaces theory (with particular stress on L^2, l^2 and abstract Fouries series on orthonormal bases). Taking into account the small price, this book is really a "must" for every mathematician and theoretical physicist: it is an unsurpassed introduction to the matter; clearly it doesn't cover more advanced parts of the theory.Explanations are very clear (unfortunately the page layout does not help); the exercises are helpful.In addition to a "linear" and "tidy" exposition of the general theory and the principal results, the book contains a lot of in-depth material, applications to parts of Mathematics not directly linked to functional analysis: as only great masters and experts can do, the author does not disdain to leave sometimes the general theory path to show the reader connections and applications of theorems and concepts to other areas, which the student is hardly enough "mature" to think of by himself (for instance, he illustrates the applications of contraction mapping principle - also known as Banach-Caccioppoli theorem - to the existence-and-uniqueness theorems for ordinary differential equations and linear equations systems, and makes use of it to demonstrate Neumann theorem on the operators geometric series, and also to solve Fredholm and Volterra integral equations: a collection of extraordinary "delicacies"). ⭐Il libro è un classico tuttora validissimo, nulla da dire... Ma la versione Kindle è una presa in giro: incompleta, una scelta di pagine fotocopiate impaginate in ordine casuale! Con Amazon non mi era mai capitata una cosa del genere! Delusissimo...... ein wundervolles Buch von wunderbaren Mathematikern.Eignet sich super für das Studium oder als Nachschlagewerk.Empfehlenswert. Würde ich nochmal kaufen.The format of the content is totally broken. I can not read it at all. It's so sad because the the book itself is so good as a textbook in this field. ⭐Clear and easy to understand even for a non native english speaker. I found it quite useful during my second year of studies in physics.

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