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Author: Stefan Cobzas
Publisher: Springer Science & Business Media
ISBN: 3034804784
Category : Mathematics
Languages : en
Pages : 229
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Book Description
An asymmetric norm is a positive definite sublinear functional p on a real vector space X. The topology generated by the asymmetric norm p is translation invariant so that the addition is continuous, but the asymmetry of the norm implies that the multiplication by scalars is continuous only when restricted to non-negative entries in the first argument. The asymmetric dual of X, meaning the set of all real-valued upper semi-continuous linear functionals on X, is merely a convex cone in the vector space of all linear functionals on X. In spite of these differences, many results from classical functional analysis have their counterparts in the asymmetric case, by taking care of the interplay between the asymmetric norm p and its conjugate. Among the positive results one can mention: Hahn–Banach type theorems and separation results for convex sets, Krein–Milman type theorems, analogs of the fundamental principles – open mapping, closed graph and uniform boundedness theorems – an analog of the Schauder’s theorem on the compactness of the conjugate mapping. Applications are given to best approximation problems and, as relevant examples, one considers normed lattices equipped with asymmetric norms and spaces of semi-Lipschitz functions on quasi-metric spaces. Since the basic topological tools come from quasi-metric spaces and quasi-uniform spaces, the first chapter of the book contains a detailed presentation of some basic results from the theory of these spaces. The focus is on results which are most used in functional analysis – completeness, compactness and Baire category – which drastically differ from those in metric or uniform spaces. The book is fairly self-contained, the prerequisites being the acquaintance with the basic results in topology and functional analysis, so it may be used for an introduction to the subject. Since new results, in the focus of current research, are also included, researchers in the area can use it as a reference text.
Author: Stefan Cobzas
Publisher: Springer Science & Business Media
ISBN: 3034804784
Category : Mathematics
Languages : en
Pages : 229
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Book Description
An asymmetric norm is a positive definite sublinear functional p on a real vector space X. The topology generated by the asymmetric norm p is translation invariant so that the addition is continuous, but the asymmetry of the norm implies that the multiplication by scalars is continuous only when restricted to non-negative entries in the first argument. The asymmetric dual of X, meaning the set of all real-valued upper semi-continuous linear functionals on X, is merely a convex cone in the vector space of all linear functionals on X. In spite of these differences, many results from classical functional analysis have their counterparts in the asymmetric case, by taking care of the interplay between the asymmetric norm p and its conjugate. Among the positive results one can mention: Hahn–Banach type theorems and separation results for convex sets, Krein–Milman type theorems, analogs of the fundamental principles – open mapping, closed graph and uniform boundedness theorems – an analog of the Schauder’s theorem on the compactness of the conjugate mapping. Applications are given to best approximation problems and, as relevant examples, one considers normed lattices equipped with asymmetric norms and spaces of semi-Lipschitz functions on quasi-metric spaces. Since the basic topological tools come from quasi-metric spaces and quasi-uniform spaces, the first chapter of the book contains a detailed presentation of some basic results from the theory of these spaces. The focus is on results which are most used in functional analysis – completeness, compactness and Baire category – which drastically differ from those in metric or uniform spaces. The book is fairly self-contained, the prerequisites being the acquaintance with the basic results in topology and functional analysis, so it may be used for an introduction to the subject. Since new results, in the focus of current research, are also included, researchers in the area can use it as a reference text.
Author: Andreea Fulga
Publisher: MDPI
ISBN: 3039288814
Category : Mathematics
Languages : en
Pages : 102
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Book Description
Metric fixed-point theory lies in the intersection of three main subjects: topology, functional analysis, and applied mathematics. The first fixed-point theorem, also known as contraction mapping principle, was abstracted by Banach from the papers of Liouville and Picard, in which certain differential equations were solved by using the method of successive approximation. In other words, fixed-point theory developed from applied mathematics and has developed in functional analysis and topology. Fixed-point theory is a dynamic research subject that has never lost the attention of researchers, as it is very open to development both in theoretical and practical fields. In this Special Issue, among several submissions, we selected eight papers that we believe will be interesting to researchers who study metric fixed-point theory and related applications. It is great to see that this Special Issue fulfilled its aims. There are not only theoretical results but also some applications that were based on obtained fixed-point results. In addition, the presented results have great potential to be improved, extended, and generalized in distinct ways. The published results also have a wide application potential in various qualitative sciences, including physics, economics, computer science, engineering, and so on.
Author: Mark Grigorʹevich Kreĭn
Publisher: American Mathematical Soc.
ISBN: 9780821886717
Category : Mathematics
Languages : en
Pages : 430
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Book Description
In this book, an extensive circle of questions originating in the classical work of P. L. Chebyshev and A. A. Markov is considered from the more modern point of view. It is shown how results and methods of the generalized moment problem are interlaced with various questions of the geometry of convex bodies, algebra, and function theory. From this standpoint, the structure of convex and conical hulls of curves is studied in detail and isoperimetric inequalities for convex hulls are established; a theory of orthogonal and quasiorthogonal polynomials is constructed; problems on limiting values of integrals and on least deviating functions (in various metrics) are generalized and solved; problems in approximation theory and interpolation and extrapolation in various function classes (analytic, absolutely monotone, almost periodic, etc.) are solved, as well as certain problems in optimal control of linear objects.
Author: Mark Burgin
Publisher: CRC Press
ISBN: 1351800299
Category : Mathematics
Languages : en
Pages : 324
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Book Description
This new volume shows how it is possible to further develop and essentially extend the theory of operators in infinite-dimensional vector spaces, which plays an important role in mathematics, physics, information theory, and control theory. The book describes new mathematical structures, such as hypernorms, hyperseminorms, hypermetrics, semitopological vector spaces, hypernormed vector spaces, and hyperseminormed vector spaces. It develops mathematical tools for the further development of functional analysis and broadening of its applications. Exploration of semitopological vector spaces, hypernormed vector spaces, hyperseminormed vector spaces, and hypermetric vector spaces is the main topic of this book. A new direction in functional analysis, called quantum functional analysis, has been developed based on polinormed and multinormed vector spaces and linear algebras. At the same time, normed vector spaces and topological vector spaces play an important role in physics and in control theory. To make this book comprehendible for the reader and more suitable for students with some basic knowledge in mathematics, denotations and definitions of the main mathematical concepts and structures used in the book are included in the appendix, making the book useful for enhancing traditional courses of calculus for undergraduates, as well as for separate courses for graduate students. The material of Semitopological Vector Spaces: Hypernorms, Hyperseminorms and Operators is closely related to what is taught at colleges and universities. It is possible to use a definite number of statements from the book as exercises for students because their proofs are not given in the book but left for the reader.
Author: Yury Evtushenko
Publisher: Springer
ISBN: 3030109348
Category : Computers
Languages : en
Pages : 538
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Book Description
This book constitutes the refereed proceedings of the 9th International Conference on Optimization and Applications, OPTIMA 2018, held in Petrovac, Montenegro, in October 2018.The 35 revised full papers and the one short paper presented were carefully reviewed and selected from 103 submissions. The papers are organized in topical sections on mathematical programming; combinatorial and discrete optimization; optimal control; optimization in economy, finance and social sciences; applications.
Author: Tom Richmond
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110686570
Category : Mathematics
Languages : en
Pages : 326
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Book Description
The first half of the book provides an introduction to general topology, with ample space given to exercises and carefully selected applications. The second half of the text includes topics in asymmetric topology, a field motivated by applications in computer science. Recurring themes include the interactions of topology with order theory and mathematics designed to model loss-of-resolution situations.
Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 476
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Book Description
Author: L. V. Kantorovich
Publisher: Elsevier
ISBN: 1483138259
Category : Mathematics
Languages : en
Pages : 605
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Book Description
Functional Analysis, Second Edition is an exposition of the theory of topological vector spaces, partially ordered spaces, and the development of the theory of integral operators and their representations on ideal spaces of measurable functions. Although this edition has deviated substantially from the first edition, it has still retained the overall plan, selection, and arrangement of the topics. The text is primarily devoted to the applications of functional analysis to applied analysis. However, these concepts have been extended and modernized. Some topics of functional analysis connected with applications to mathematical economics and control theory are also included in this edition. The applications of functional analysis are both wide and far-reaching as these are common language for all areas of mathematics involving the concept of continuity. Those who are in the field of mathematics, mechanics, and theoretical physics will find this book a valuable resource.
Author: Ştefan Cobzaş
Publisher: Springer
ISBN: 3030164896
Category : Mathematics
Languages : en
Pages : 605
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Book Description
The aim of this book is to present various facets of the theory and applications of Lipschitz functions, starting with classical and culminating with some recent results. Among the included topics we mention: characterizations of Lipschitz functions and relations with other classes of functions, extension results for Lipschitz functions and Lipschitz partitions of unity, Lipschitz free Banach spaces and their applications, compactness properties of Lipschitz operators, Bishop-Phelps type results for Lipschitz functionals, applications to best approximation in metric and in metric linear spaces, Kantorovich-Rubinstein norm and applications to duality in the optimal transport problem, Lipschitz mappings on geodesic spaces. The prerequisites are basic results in real analysis, functional analysis, measure theory (including vector measures) and topology, which, for reader's convenience, are surveyed in the first chapter of the book.
Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 800
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Book Description
