Vitali Milman's Books

Functional Analysis

An Introduction

By: Yuli Eidelman,Vitali D. Milman,Antonis Tsolomitis

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Functional Analysis

An Introduction

By: Yuli Eidelman,Vitali D. Milman,Antonis Tsolomitis

Introduces the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators and spectral theory of self-adjoint operators. This work presents the theorems and methods of abstract functional analysis and applications of these methods to Banach algebras and theory of unbounded self-adjoint operators.

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344

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821836463

ISBN 13

9780821836460

Asymptotic Geometric Analysis, Part II

By: Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman

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Asymptotic Geometric Analysis, Part II

By: Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman

This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as volume 202 in this series. Asymptotic geometric analysis studies properties of geometric objects, such as normed spaces, convex bodies, or convex functions, when the dimensions of these objects increase to infinity. The asymptotic approach reveals many very novel phenomena which influence other fields in mathematics, especially where a large data set is of main concern, or a number of parameters which becomes uncontrollably large. One of the important features of this new theory is in developing tools which allow studying high parametric families. Among the topics covered in the book are measure concentration, isoperimetric constants of log-concave measures, thin-shell estimates, stochastic localization, the geometry of Gaussian measures, volume inequalities for convex bodies, local theory of Banach spaces, type and cotype, the Banach-Mazur compactum, symmetrizations, restricted invertibility, and functional versions of geometric notions and inequalities.

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645

Publisher

American Mathematical Society

Published Date

ISBN 10

1470463601

ISBN 13

9781470463601

Asymptotic Geometric Analysis, Part I

By: Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman

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Asymptotic Geometric Analysis, Part I

By: Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman

The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an "isomorphic" point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here in a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the "isomorphic isoperimetric inequalities" which led to the discovery of the "concentration phenomenon", one of the most powerful tools of the theory, responsible for many counterintuitive results. A central theme in this book is the interaction of randomness and pattern. At first glance, life in high dimension seems to mean the existence of multiple "possibilities", so one may expect an increase in the diversity and complexity as dimension increases. However, the concentration of measure and effects caused by convexity show that this diversity is compensated and order and patterns are created for arbitrary convex bodies in the mixture caused by high dimensionality. The book is intended for graduate students and researchers who want to learn about this exciting subject. Among the topics covered in the book are convexity, concentration phenomena, covering numbers, Dvoretzky-type theorems, volume distribution in convex bodies, and more.

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Page Count

473

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470421933

ISBN 13

9781470421939

Asymptotic Theory of Finite Dimensional Normed Spaces

Isoperimetric Inequalities in Riemannian Manifolds

By: Vitali D. Milman,Gideon Schechtman

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Asymptotic Theory of Finite Dimensional Normed Spaces

Isoperimetric Inequalities in Riemannian Manifolds

By: Vitali D. Milman,Gideon Schechtman

This book deals with the geometrical structure of finite dimensional normed spaces, as the dimension grows to infinity. This is a part of what came to be known as the Local Theory of Banach Spaces (this name was derived from the fact that in its first stages, this theory dealt mainly with relating the structure of infinite dimensional Banach spaces to the structure of their lattice of finite dimensional subspaces). Our purpose in this book is to introduce the reader to some of the results, problems, and mainly methods developed in the Local Theory, in the last few years. This by no means is a complete survey of this wide area. Some of the main topics we do not discuss here are mentioned in the Notes and Remarks section. Several books appeared recently or are going to appear shortly, which cover much of the material not covered in this book. Among these are Pisier's [Pis6] where factorization theorems related to Grothendieck's theorem are extensively discussed, and Tomczak-Jaegermann's [T-Jl] where operator ideals and distances between finite dimensional normed spaces are studied in detail. Another related book is Pietch's [Pie].

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Page Count

166

Publisher

Springer

Published Date

ISBN 10

3540388222

ISBN 13

9783540388227

Operator Relations Characterizing Derivatives

By: Hermann König,Vitali Milman

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Operator Relations Characterizing Derivatives

By: Hermann König,Vitali Milman

This monograph develops an operator viewpoint for functional equations in classical function spaces of analysis, thus filling a void in the mathematical literature. Major constructions or operations in analysis are often characterized by some elementary properties, relations or equations which they satisfy. The authors present recent results on the problem to what extent the derivative is characterized by equations such as the Leibniz rule or the Chain rule operator equation in Ck-spaces. By localization, these operator equations turn into specific functional equations which the authors then solve. The second derivative, Sturm-Liouville operators and the Laplacian motivate the study of certain "second-order" operator equations. Additionally, the authors determine the general solution of these operator equations under weak assumptions of non-degeneration. In their approach, operators are not required to be linear, and the authors also try to avoid continuity conditions. The Leibniz rule, the Chain rule and its extensions turn out to be stable under perturbations and relaxations of assumptions on the form of the operators. The results yield an algebraic understanding of first- and second-order differential operators. Because the authors have chosen to characterize the derivative by algebraic relations, the rich operator-type structure behind the fundamental notion of the derivative and its relatives in analysis is discovered and explored. The book does not require any specific knowledge of functional equations. All needed results are presented and proven and the book is addressed to a general mathematical audience.

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193

Publisher

Springer

Published Date

ISBN 10

3030002411

ISBN 13

9783030002411

Functional Analysis

An Introduction

By: Yuli Eidelman,Vitali D. Milman,Antonis Tsolomitis

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Functional Analysis

An Introduction

By: Yuli Eidelman,Vitali D. Milman,Antonis Tsolomitis

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Page Count

344

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821836463

ISBN 13

9780821836460

Operator Relations Characterizing Derivatives

By: Hermann König,Vitali Milman

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Operator Relations Characterizing Derivatives

By: Hermann König,Vitali Milman

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193

Publisher

Springer

Published Date

ISBN 10

3030002411

ISBN 13

9783030002411

Asymptotic Theory of Finite Dimensional Normed Spaces

Isoperimetric Inequalities in Riemannian Manifolds

By: Vitali D. Milman,Gideon Schechtman

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Asymptotic Theory of Finite Dimensional Normed Spaces

Isoperimetric Inequalities in Riemannian Manifolds

By: Vitali D. Milman,Gideon Schechtman

Vol. 1200 of the LNM series deals with the geometrical structure of finite dimensional normed spaces. One of the main topics is the estimation of the dimensions of euclidean and l^n p spaces which nicely embed into diverse finite-dimensional normed spaces. An essential method here is the concentration of measure phenomenon which is closely related to large deviation inequalities in Probability on the one hand, and to isoperimetric inequalities in Geometry on the other. The book contains also an appendix, written by M. Gromov, which is an introduction to isoperimetric inequalities on riemannian manifolds. Only basic knowledge of Functional Analysis and Probability is expected of the reader. The book can be used (and was used by the authors) as a text for a first or second graduate course. The methods used here have been useful also in areas other than Functional Analysis (notably, Combinatorics).

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Page Count

166

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3540167692

ISBN 13

9783540167693

Asymptotic Geometric Analysis, Part II

By: Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman

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Asymptotic Geometric Analysis, Part II

By: Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman

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Page Count

645

Publisher

American Mathematical Society

Published Date

ISBN 10

1470463601

ISBN 13

9781470463601

Geometric Aspects of Functional Analysis

Israel Seminar 2006–2010

By: Bo'az Klartag,Shahar Mendelson,Vitali D. Milman

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Geometric Aspects of Functional Analysis

Israel Seminar 2006–2010

By: Bo'az Klartag,Shahar Mendelson,Vitali D. Milman

This collection of original papers related to the Israeli GAFA seminar (on Geometric Aspects of Functional Analysis) from the years 2006 to 2011 continues the long tradition of the previous volumes, which reflect the general trends of Asymptotic Geometric Analysis, understood in a broad sense, and are a source of inspiration for new research. Most of the papers deal with various aspects of the theory, including classical topics in the geometry of convex bodies, inequalities involving volumes of such bodies or more generally, logarithmically-concave measures, valuation theory, probabilistic and isoperimetric problems in the combinatorial setting, volume distribution on high-dimensional spaces and characterization of classical constructions in Geometry and Analysis (like the Legendre and Fourier transforms, derivation and others). All the papers here are original research papers.

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Page Count

449

Publisher

Springer

Published Date

ISBN 10

3642298486

ISBN 13

9783642298486

Book's by Vitali Milman

Functional Analysis

Functional Analysis

Asymptotic Geometric Analysis, Part II

Asymptotic Geometric Analysis, Part II

Asymptotic Geometric Analysis, Part I

Asymptotic Geometric Analysis, Part I

Asymptotic Theory of Finite Dimensional Normed Spaces

Asymptotic Theory of Finite Dimensional Normed Spaces

Operator Relations Characterizing Derivatives

Operator Relations Characterizing Derivatives

Functional Analysis

Functional Analysis

Operator Relations Characterizing Derivatives

Operator Relations Characterizing Derivatives

Asymptotic Theory of Finite Dimensional Normed Spaces

Asymptotic Theory of Finite Dimensional Normed Spaces

Asymptotic Geometric Analysis, Part II

Asymptotic Geometric Analysis, Part II

Geometric Aspects of Functional Analysis

Geometric Aspects of Functional Analysis

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