Download or read book The Representations of Linear Functionals by Measures on Sets of Extreme Points written by Errett Bishop and published by . This book was released on 1959 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Selected Papers Of Errett Bishop written by and published by World Scientific. This book was released on 1986-12-01 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: Errett Bishop's mathematical work was divided between complex and functional analysis, and constructive mathematics. The influence of his discoveries in these areas is still strongly felt today.
Download or read book Handbook of Measure Theory written by E. Pap and published by Elsevier. This book was released on 2002-10-31 with total page 1632 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main goal of this Handbook is to survey measure theory with its many different branches and its relations with other areas of mathematics. Mostly aggregating many classical branches of measure theory the aim of the Handbook is also to cover new fields, approaches and applications which support the idea of "measure" in a wider sense, e.g. the ninth part of the Handbook. Although chapters are written of surveys in the various areas they contain many special topics and challenging problems valuable for experts and rich sources of inspiration. Mathematicians from other areas as well as physicists, computer scientists, engineers and econometrists will find useful results and powerful methods for their research. The reader may find in the Handbook many close relations to other mathematical areas: real analysis, probability theory, statistics, ergodic theory, functional analysis, potential theory, topology, set theory, geometry, differential equations, optimization, variational analysis, decision making and others. The Handbook is a rich source of relevant references to articles, books and lecture notes and it contains for the reader's convenience an extensive subject and author index.
Download or read book Vector Measures written by Joseph Diestel and published by American Mathematical Soc.. This book was released on 1977-06-01 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this survey the authors endeavor to give a comprehensive examination of the theory of measures having values in Banach spaces. The interplay between topological and geometric properties of Banach spaces and the properties of measures having values in Banach spaces is the unifying theme. The first chapter deals with countably additive vector measures finitely additive vector measures, the Orlicz-Pettis theorem and its relatives. Chapter II concentrates on measurable vector valued functions and the Bochner integral. Chapter III begins the study of the interplay among the Radon-Nikodym theorem for vector measures, operators on $L_1$ and topological properties of Banach spaces. A variety of applications is given in the next chapter. Chapter V deals with martingales of Bochner integrable functions and their relation to dentable subsets of Banach spaces. Chapter VI is devoted to a measure-theoretic study of weakly compact absolutely summing and nuclear operators on spaces of continuous functions. In Chapter VII a detailed study of the geometry of Banach spaces with the Radon-Nikodym property is given. The next chapter deals with the use of Radon-Nikodym theorems in the study of tensor products of Banach spaces. The last chapter concludes the survey with a discussion of the Liapounoff convexity theorem and other geometric properties of the range of a vector measure. Accompanying each chapter is an extensive survey of the literature and open problems.
Download or read book Compact Convex Sets and Boundary Integrals written by Erik M. Alfsen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: The importance of convexity arguments in functional analysis has long been realized, but a comprehensive theory of infinite-dimensional convex sets has hardly existed for more than a decade. In fact, the integral representation theorems of Choquet and Bishop -de Leeuw together with the uniqueness theorem of Choquet inaugurated a new epoch in infinite-dimensional convexity. Initially considered curious and tech nically difficult, these theorems attracted many mathematicians, and the proofs were gradually simplified and fitted into a general theory. The results can no longer be considered very "deep" or difficult, but they certainly remain all the more important. Today Choquet Theory provides a unified approach to integral representations in fields as diverse as potential theory, probability, function algebras, operator theory, group representations and ergodic theory. At the same time the new concepts and results have made it possible, and relevant, to ask new questions within the abstract theory itself. Such questions pertain to the interplay between compact convex sets K and their associated spaces A(K) of continuous affine functions; to the duality between faces of K and appropriate ideals of A(K); to dominated extension problems for continuous affine functions on faces; and to direct convex sum decomposition into faces, as well as to integral for mulas generalizing such decompositions. These problems are of geometric interest in their own right, but they are primarily suggested by applica tions, in particular to operator theory and function algebras.
Download or read book Geometric Aspects of Convex Sets with the Radon Nikodym Property written by R. D. Bourgin and published by Springer. This book was released on 2006-11-15 with total page 485 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Uniform Algebras and Jensen Measures written by Theodore W. Gamelin and published by Cambridge University Press. This book was released on 1978 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes are devoted to an area of current research interest that bridges functional analysis and function theory. The unifying theme is the notion of subharmonicity with respect to a uniform algebra. The topics covered include the rudiments of Choquet theory, various classes of representing measures, the duality between abstract sub-harmonic functions and Jensen measures, applications to problems of approximation of plurisubharmonic functions of several complex variables, and Cole's theory of estimates for conjugate functions. Many of the results are published here for the first time in monograph form.
Download or read book Functional Analysis written by Edward W. Odell and published by Springer. This book was released on 2006-11-14 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this volume are based on talks given in a seminar at Austin during 1986-87. They range from those dealing with fresh research and discoveries to exposition and new proofs of older results. The main topics and themes include geometric and analytic properties of infinite-dimensional Banach spaces and their convex subsets as well as some aspects of Banach spaces associated with harmonic analysis and Banach algebras.
Download or read book Operator Valued Measures and Integrals for Cone Valued Functions written by Walter Roth and published by Springer. This book was released on 2008-12-17 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Integration theory deals with extended real-valued, vector-valued, or operator-valued measures and functions. Different approaches are applied in each of these cases using different techniques. The order structure of the (extended) real number system is used for real-valued functions and measures whereas suprema and infima are replaced with topological limits in the vector-valued case. A novel approach employing more general structures, locally convex cones, which are natural generalizations of locally convex vector spaces, is introduced here. This setting allows developing a general theory of integration which simultaneously deals with all of the above-mentioned cases.
Download or read book Functional Analysis Surveys and Recent Results II written by K.-D. Bierstedt and published by Elsevier. This book was released on 1980-01-01 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: Functional Analysis: Surveys and Recent Results II
Download or read book The Selected Works of V S Varadarajan written by V. S. Varadarajan and published by American Mathematical Soc.. This book was released on 1999 with total page 652 pages. Available in PDF, EPUB and Kindle. Book excerpt: V.S. Varadarajan has made significant contributions to a remarkably broad range of mathematical subjects which include probability theory, various mathematical aspects of quantum mechanics, harmonic analysis on reductive groups and symmetric spaces, and the modern theory of meromorphic differential equations. The papers included in this volume have been selected to highlight these contributions. This book is jointly published by the AMS and the International Press.
Download or read book Integral Representation written by Walter Roth and published by Walter de Gruyter GmbH & Co KG. This book was released on 2023-10-02 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Lower Previsions written by Matthias C. M. Troffaes and published by John Wiley & Sons. This book was released on 2014-04-09 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book has two main purposes. On the one hand, it provides a concise and systematic development of the theory of lower previsions, based on the concept of acceptability, in spirit of the work of Williams and Walley. On the other hand, it also extends this theory to deal with unbounded quantities, which abound in practical applications. Following Williams, we start out with sets of acceptable gambles. From those, we derive rationality criteria---avoiding sure loss and coherence---and inference methods---natural extension---for (unconditional) lower previsions. We then proceed to study various aspects of the resulting theory, including the concept of expectation (linear previsions), limits, vacuous models, classical propositional logic, lower oscillations, and monotone convergence. We discuss n-monotonicity for lower previsions, and relate lower previsions with Choquet integration, belief functions, random sets, possibility measures, various integrals, symmetry, and representation theorems based on the Bishop-De Leeuw theorem. Next, we extend the framework of sets of acceptable gambles to consider also unbounded quantities. As before, we again derive rationality criteria and inference methods for lower previsions, this time also allowing for conditioning. We apply this theory to construct extensions of lower previsions from bounded random quantities to a larger set of random quantities, based on ideas borrowed from the theory of Dunford integration. A first step is to extend a lower prevision to random quantities that are bounded on the complement of a null set (essentially bounded random quantities). This extension is achieved by a natural extension procedure that can be motivated by a rationality axiom stating that adding null random quantities does not affect acceptability. In a further step, we approximate unbounded random quantities by a sequences of bounded ones, and, in essence, we identify those for which the induced lower prevision limit does not depend on the details of the approximation. We call those random quantities 'previsible'. We study previsibility by cut sequences, and arrive at a simple sufficient condition. For the 2-monotone case, we establish a Choquet integral representation for the extension. For the general case, we prove that the extension can always be written as an envelope of Dunford integrals. We end with some examples of the theory.
Download or read book Integral Representation Theory written by Jaroslav Lukeš and published by Walter de Gruyter. This book was released on 2010 with total page 732 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents the state of the art of convexity, with an emphasis to integral representation. The exposition is focused on Choquet's theory of function spaces with a link to compact convex sets. An important feature of the book is an interplay between various mathematical subjects, such as functional analysis, measure theory, descriptive set theory, Banach spaces theory and potential theory. A substantial part of the material is of fairly recent origin and many results appear in the book form for the first time. The text is self-contained and covers a wide range of applications. From the contents: Geometry of convex sets Choquet theory of function spaces Affine functions on compact convex sets Perfect classes of functions and representation of affine functions Simplicial function spaces Choquet's theory of function cones Topologies on boundaries Several results on function spaces and compact convex sets Continuous and measurable selectors Construction of function spaces Function spaces in potential theory and Dirichlet problem Applications
Download or read book Diffusion Processes and their Sample Paths written by Kiyosi Itô and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since its first publication in 1965 in the series Grundlehren der mathematischen Wissenschaften this book has had a profound and enduring influence on research into the stochastic processes associated with diffusion phenomena. Generations of mathematicians have appreciated the clarity of the descriptions given of one- or more- dimensional diffusion processes and the mathematical insight provided into Brownian motion. Now, with its republication in the Classics in Mathematics it is hoped that a new generation will be able to enjoy the classic text of Itô and McKean.
Download or read book Handbook of the Geometry of Banach Spaces written by and published by Elsevier. This book was released on 2001-08-15 with total page 1017 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Handbook presents an overview of most aspects of modernBanach space theory and its applications. The up-to-date surveys, authored by leading research workers in the area, are written to be accessible to a wide audience. In addition to presenting the state of the art of Banach space theory, the surveys discuss the relation of the subject with such areas as harmonic analysis, complex analysis, classical convexity, probability theory, operator theory, combinatorics, logic, geometric measure theory, and partial differential equations. The Handbook begins with a chapter on basic concepts in Banachspace theory which contains all the background needed for reading any other chapter in the Handbook. Each of the twenty one articles in this volume after the basic concepts chapter is devoted to one specific direction of Banach space theory or its applications. Each article contains a motivated introduction as well as an exposition of the main results, methods, and open problems in its specific direction. Most have an extensive bibliography. Many articles contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers. As well as being valuable to experienced researchers in Banach space theory, the Handbook should be an outstanding source for inspiration and information to graduate students and beginning researchers. The Handbook will be useful for mathematicians who want to get an idea of the various developments in Banach space theory.
Download or read book Unbounded Operator Algebras and Representation Theory written by K. Schmüdgen and published by Birkhäuser. This book was released on 2013-11-11 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: *-algebras of unbounded operators in Hilbert space, or more generally algebraic systems of unbounded operators, occur in a natural way in unitary representation theory of Lie groups and in the Wightman formulation of quantum field theory. In representation theory they appear as the images of the associated representations of the Lie algebras or of the enveloping algebras on the Garding domain and in quantum field theory they occur as the vector space of field operators or the *-algebra generated by them. Some of the basic tools for the general theory were first introduced and used in these fields. For instance, the notion of the weak (bounded) commutant which plays a fundamental role in thegeneraltheory had already appeared in quantum field theory early in the six ties. Nevertheless, a systematic study of unbounded operator algebras began only at the beginning of the seventies. It was initiated by (in alphabetic order) BORCHERS, LASSNER, POWERS, UHLMANN and VASILIEV. J1'rom the very beginning, and still today, represen tation theory of Lie groups and Lie algebras and quantum field theory have been primary sources of motivation and also of examples. However, the general theory of unbounded operator algebras has also had points of contact with several other disciplines. In particu lar, the theory of locally convex spaces, the theory of von Neumann algebras, distri bution theory, single operator theory, the momcnt problem and its non-commutative generalizations and noncommutative probability theory, all have interacted with our subject.
