metric spaces springer undergraduate mathematics series

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Metric Spaces

Author : Mícheál O'Searcoid
ISBN : 1846286271
Genre : Mathematics
File Size : 80. 58 MB
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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.

Metric Spaces

Author : Mícheál O'Searcoid
ISBN : 184800494X
Genre : Mathematics
File Size : 32. 37 MB
Format : PDF
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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.

Metric Spaces

Author : Mícheál O'Searcoid
ISBN : 1846283698
Genre : Mathematics
File Size : 76. 51 MB
Format : PDF
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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.

Set Theory And Metric Spaces

Author : Irving Kaplansky
ISBN : 9780821826942
Genre : Mathematics
File Size : 35. 68 MB
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This is a book that could profitably be read by many graduate students or by seniors in strong major programs ... has a number of good features. There are many informal comments scattered between the formal development of theorems and these are done in a light and pleasant style. ... There is a complete proof of the equivalence of the axiom of choice, Zorn's Lemma, and well-ordering, as well as a discussion of the use of these concepts. There is also an interesting discussion of the continuum problem ... The presentation of metric spaces before topological spaces ... should be welcomed by most students, since metric spaces are much closer to the ideas of Euclidean spaces with which they are already familiar. --Canadian Mathematical Bulletin Kaplansky has a well-deserved reputation for his expository talents. The selection of topics is excellent. --Lance Small, UC San Diego This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.

Essential Real Analysis

Author : Michael Field
ISBN : 9783319675466
Genre : Mathematics
File Size : 43. 72 MB
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This book provides a rigorous introduction to the techniques and results of real analysis, metric spaces and multivariate differentiation, suitable for undergraduate courses. Starting from the very foundations of analysis, it offers a complete first course in real analysis, including topics rarely found in such detail in an undergraduate textbook such as the construction of non-analytic smooth functions, applications of the Euler-Maclaurin formula to estimates, and fractal geometry. Drawing on the author’s extensive teaching and research experience, the exposition is guided by carefully chosen examples and counter-examples, with the emphasis placed on the key ideas underlying the theory. Much of the content is informed by its applicability: Fourier analysis is developed to the point where it can be rigorously applied to partial differential equations or computation, and the theory of metric spaces includes applications to ordinary differential equations and fractals. Essential Real Analysis will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty, including some suitable for group work or class discussion, make this book suitable for self-study as well as lecture courses.

Metric Spaces

Author : Victor Bryant
ISBN : 0521318971
Genre : Mathematics
File Size : 22. 71 MB
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An introduction to metric spaces for those interested in the applications as well as theory.

Linear Functional Analysis

Author : Bryan Rynne
ISBN : 9781447136552
Genre : Mathematics
File Size : 22. 6 MB
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This book provides an introduction to the ideas and methods of linear func tional analysis at a level appropriate to the final year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the the ory of metric spaces). Part of the development of functional analysis can be traced to attempts to find a suitable framework in which to discuss differential and integral equa tions. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions defined on some set. In general, such a vector space is infinite-dimensional. This leads to difficulties in that, although many of the elementary properties of finite-dimensional vector spaces hold in infinite dimensional vector spaces, many others do not. For example, in general infinite dimensional vector spaces there is no framework in which to make sense of an alytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now briefly outline the contents of the book.

Elements Of Abstract Analysis

Author : Mícheál O'Searcoid
ISBN : 9781447101796
Genre : Mathematics
File Size : 56. 28 MB
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While there are many books on functional analysis, Elements of Abstract Analysis takes a very different approach. Unlike other books, it provides a comprehensive overview of the elementary concepts of analysis while preparing students to cross the threshold of functional analysis. The book is written specifically for final-year undergraduate students who should already be familiar with most of the mathematical structures discussed. It reviews the concepts at a slightly greater level of abstraction and enables students to understand their place within the broad framework of set-based mathematics. The book has been clearly written and contains numerous exercises and examples, making it an a rigorous and self-contained introductory text on functional analysis.

Metric Spaces

Author : Satish Shirali
ISBN : 1852339225
Genre : Mathematics
File Size : 35. 1 MB
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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.

Topological Spaces

Author : Gerard Buskes
ISBN : 0387949941
Genre : Mathematics
File Size : 57. 3 MB
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gentle introduction to the subject, leading the reader to understand the notion of what is important in topology with regard to geometry. Divided into three sections - The line and the plane, Metric spaces and Topological spaces -, the book eases the move into higher levels of abstraction. Students are thereby informally assisted in learning new ideas while remaining on familiar territory. The authors do not assume previous knowledge of axiomatic approach or set theory. Similarly, they have restricted the mathematical vocabulary in the book so as to avoid overwhelming the reader, and the concept of convergence is employed to allow students to focus on a central theme while moving to a natural understanding of the notion of topology. The pace of the book is relaxed with gradual acceleration: the first nine sections form a balanced course in metric spaces for undergraduates while also containing ample material for a two-semester graduate course. Finally, the book illustrates the many connections between topology and other subjects, such as analysis and set theory, via the inclusion of "Extras" at the end of each chapter presenting a brief foray outside topology.

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