An introduction to abstract analysis
著者
書誌事項
An introduction to abstract analysis
Chapman and Hall, 1990
1st ed.
- : pbk
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内容説明・目次
- 巻冊次
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ISBN 9780412310805
内容説明
An overview of abstract analysis and the language of normed linear spaces, which lie at the heart of modern mathematics. It gives an introduction to the basic notions of convergence of sequences, continuity of functions, open and closed set, compactness, completeness and separability. It includes chapters on differentiation, linear mappings, integration, the implicit function theorem as well as exercises and examples of applications of the theory to diverse areas of mathematics. The book is intended for the student who has had a first course in mathematical analysis or advanced calculus.
目次
Basic ideas. Some simple results. Open and closed sets. Denseness, separability and completeness. Function spaces. Compactness. The contraction mapping theorem. Linear mappings. Differentiation in R2. Differetiation - a more abstract viewpoint. The Riemann integral. Two hard results.
- 巻冊次
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: pbk ISBN 9780412310904
内容説明
Abstract analysis, and particularly the language of normed linear spaces, now lies at the heart of a major portion of modern mathematics. Unfortunately, it is also a subject which students seem to find quite challenging and difficult. This book presumes that the student has had a first course in mathematical analysis or advanced calculus, but it does not presume the student has achieved mastery of such a course. Accordingly, a gentle introduction to the basic notions of convergence of sequences, continuity of functions, open and closed set, compactness, completeness and separability is given.
The pace in the early chapters does not presume in any way that the readers have at their fingertips the techniques provided by an introductory course. Instead, considerable care is taken to introduce and use the basic methods of proof in a slow and explicit fashion. As the chapters progress, the pace does quicken and later chapters on differentiation, linear mappings, integration and the implicit function theorem delve quite deeply into interesting mathematical areas.
There are many exercises and many examples of applications of the theory to diverse areas of mathematics. Some of these applications take considerable space and time to develop, and make interesting reading in their own right.
The treatment of the subject is deliberately not a comprehensive one. The aim is to convince the undergraduate reader that analysis is a stimulating, useful, powerful and comprehensible tool in modern mathematics. This book will whet the readers' appetite, not overwhelm them with material.
目次
Basic ideas. Some simple results. Open and closed sets. Denseness, separability and completeness. Function spaces. Compactness. The contraction mapping theorem. Linear mappings. Differentiation in IR squared. Differentiation - a more abstract viewpoint. The Riemann integral. Two hard results. Appendix. Countability.
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