**Author**: John R. Giles

**Publisher:** Cambridge University Press

**ISBN:** 9780521359283

**Category : **Mathematics

**Languages : **en

**Pages : **257

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**Book Description**

Assuming a basic knowledge of real analysis and linear algebra, the student is given some familiarity with the axiomatic method in analysis and is shown the power of this method in exploiting the fundamental analysis structures underlying a variety of applications. Although the text is titled metric spaces, normed linear spaces are introduced immediately because this added structure is present in many examples and its recognition brings an interesting link with linear algebra; finite dimensional spaces are discussed earlier. It is intended that metric spaces be studied in some detail before general topology is begun. This follows the teaching principle of proceeding from the concrete to the more abstract. Graded exercises are provided at the end of each section and in each set the earlier exercises are designed to assist in the detection of the abstract structural properties in concrete examples while the latter are more conceptually sophisticated.

**Author**: John R. Giles

**Publisher:** Cambridge University Press

**ISBN:** 9780521359283

**Category : **Mathematics

**Languages : **en

**Pages : **257

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**Book Description**
Assuming a basic knowledge of real analysis and linear algebra, the student is given some familiarity with the axiomatic method in analysis and is shown the power of this method in exploiting the fundamental analysis structures underlying a variety of applications. Although the text is titled metric spaces, normed linear spaces are introduced immediately because this added structure is present in many examples and its recognition brings an interesting link with linear algebra; finite dimensional spaces are discussed earlier. It is intended that metric spaces be studied in some detail before general topology is begun. This follows the teaching principle of proceeding from the concrete to the more abstract. Graded exercises are provided at the end of each section and in each set the earlier exercises are designed to assist in the detection of the abstract structural properties in concrete examples while the latter are more conceptually sophisticated.

**Author**: J. R. Giles

**Publisher:** Cambridge University Press

**ISBN:** 9780521653756

**Category : **Mathematics

**Languages : **en

**Pages : **280

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**Book Description**
This is a basic course in functional analysis for senior undergraduate and beginning postgraduate students. The reader need only be familiarity with elementary real and complex analysis, linear algebra and have studied a course in the analysis of metric spaces; knowledge of integration theory or general topology is not required. The text concerns the structural properties of normed linear spaces in general, especially associated with dual spaces and continuous linear operators on normed linear spaces. The implications of the general theory are illustrated with a great variety of example spaces.

**Author**: Dhananjay Gopal

**Publisher:** CRC Press

**ISBN:** 1000087999

**Category : **Mathematics

**Languages : **en

**Pages : **286

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**Book Description**
This book serves as a textbook for an introductory course in metric spaces for undergraduate or graduate students. The goal is to present the basics of metric spaces in a natural and intuitive way and encourage students to think geometrically while actively participating in the learning of this subject. In this book, the authors illustrated the strategy of the proofs of various theorems that motivate readers to complete them on their own. Bits of pertinent history are infused in the text, including brief biographies of some of the central players in the development of metric spaces. The textbook is divided into seven chapters that contain the main materials on metric spaces; namely, introductory concepts, completeness, compactness, connectedness, continuous functions and metric fixed point theorems with applications. Some of the noteworthy features of this book include · Diagrammatic illustrations that encourage readers to think geometrically · Focus on systematic strategy to generate ideas for the proofs of theorems · A wealth of remarks, observations along with a variety of exercises · Historical notes and brief biographies appearing throughout the text

**Author**: Tom L. Lindstrøm

**Publisher:** American Mathematical Soc.

**ISBN:** 1470440628

**Category : **Functional analysis

**Languages : **en

**Pages : **369

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**Book Description**
Spaces is a modern introduction to real analysis at the advanced undergraduate level. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are a solid understanding of calculus and linear algebra. Two introductory chapters will help students with the transition from computation-based calculus to theory-based analysis. The main topics covered are metric spaces, spaces of continuous functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although some of the topics are more advanced than what is usually found in books of this level, care is taken to present the material in a way that is suitable for the intended audience: concepts are carefully introduced and motivated, and proofs are presented in full detail. Applications to differential equations and Fourier analysis are used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their skills and understanding. The text has been tested in classes at the University of Oslo over a number of years.

**Author**: Mohamed A. Khamsi

**Publisher:** John Wiley & Sons

**ISBN:** 1118031326

**Category : **Mathematics

**Languages : **en

**Pages : **320

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**Book Description**
Presents up-to-date Banach space results. * Features an extensive bibliography for outside reading. * Provides detailed exercises that elucidate more introductorymaterial.

**Author**: Andrey Popoff

**Publisher:** CRC Press

**ISBN:** 1000572005

**Category : **Technology & Engineering

**Languages : **en

**Pages : **450

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**Book Description**
Exploring the interrelations between generalized metric spaces, lattice-ordered groups, and order statistics, the book contains a new algebraic approach to Signal Processing Theory. It describes mathematical concepts and results important in the development, analysis, and optimization of signal processing algorithms intended for various applications. The book offers a solution of large-scale Signal Processing Theory problems of increasing both signal processing efficiency under prior uncertainty conditions and signal processing rate that is provided by multiplication-free signal processing algorithms based on lattice-ordered group operations. From simple basic relationships to computer simulation, the text covers a wide range of new mathematical techniques essential for understanding the proposed signal processing algorithms developed for solving the following problems: signal parameter and spectral estimation, signal filtering, detection, classification, and resolution; array signal processing; demultiplexing and demodulation in multi-channel communication systems and multi-station networks; wavelet analysis of 1D/ 2D signals. Along with discussing mathematical aspects, each chapter presents examples illustrating operation of signal processing algorithms developed for various applications. The book helps readers understand relations between known classic and obtained results as well as recent research trends in Signal Processing Theory and its applications, providing all necessary mathematical background concerning lattice-ordered groups to prepare readers for independent work in the marked directions including more advanced research and development.

**Author**: Assistant Professor of Mathematics Juha Heinonen

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387951041

**Category : **Mathematics

**Languages : **en

**Pages : **140

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**Book Description**
Studies basic covering theorems and their applications to analysis in metric measure spaces. This book discusses Sobolev spaces emphasizing principles that are valid in larger contexts. It also presents a basic theory of quasisymmetric maps between metric spaces.

**Author**: Mícheál O'Searcoid

**Publisher:** Springer Science & Business Media

**ISBN:** 1846286271

**Category : **Mathematics

**Languages : **en

**Pages : **304

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**Book Description**
The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.

**Author**: Satish Shirali

**Publisher:** Springer Science & Business Media

**ISBN:** 9781852339227

**Category : **Mathematics

**Languages : **en

**Pages : **222

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**Book Description**
Since the last century, the postulational method and an abstract point of view have played a vital role in the development of modern mathematics. The experience gained from the earlier concrete studies of analysis point to the importance of passage to the limit. The basis of this operation is the notion of distance between any two points of the line or the complex plane. The algebraic properties of underlying sets often play no role in the development of analysis; this situation naturally leads to the study of metric spaces. The abstraction not only simplifies and elucidates mathematical ideas that recur in different guises, but also helps eco- mize the intellectual effort involved in learning them. However, such an abstract approach is likely to overlook the special features of particular mathematical developments, especially those not taken into account while forming the larger picture. Hence, the study of particular mathematical developments is hard to overemphasize. The language in which a large body of ideas and results of functional analysis are expressed is that of metric spaces. The books on functional analysis seem to go over the preliminaries of this topic far too quickly. The present authors attempt to provide a leisurely approach to the theory of metric spaces. In order to ensure that the ideas take root gradually but firmly, a large number of examples and counterexamples follow each definition. Also included are several worked examples and exercises. Applications of the theory are spread out over the entire book.

**Author**: K. G. Binmore

**Publisher:** Cambridge University Press

**ISBN:** 9780521233507

**Category : **Mathematics

**Languages : **en

**Pages : **261

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**Book Description**
This book is an introduction to the ideas from general topology that are used in elementary analysis. It is written at a level that is intended to make the bulk of the material accessible to students in the latter part of their first year of study at a university or college although students will normally meet most of the work in their second or later years. The aim has been to bridge the gap between introductory books like the author's Mathematical Analysis: A Straightforward Approach, in which carefully selected theorems are discussed at length with numerous examples, and the more advanced book on analysis, in which the author is more concerned with providing a comprehensive and elegant theory than in smoothing the ways for beginners. An attempt has been made throughout not only to prepare the ground for more advanced work, but also to revise and to illuminate the material which students will have met previously but may have not fully understood.