Beginning functional analysis开始功能分析
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作者： Karen Saxe著
出 版 社：
出版时间： 2001-12-1字数：版次： 1页数： 197印刷时间： 2001/12/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387952246包装： 精装内容简介
The unifying approach of functional analysis is to view functions as points in some abstract vector space and the differential and integral operators relating these points as linear transformations on these spaces. The author presents the basics of functional analysis with attention paid to both expository style and technical detail, while getting to interesting results as quickly as possible. The book is accessible to students who have completed first courses in linear algebra and real analysis. Topics are developed in their historical context, with accounts of the past ¿ including biographies ¿ appearing throughout the text. The book offers suggestions and references for further study, and many exercises. Karen Saxe is Associate Professor of Mathematics at Macalester College in St. Paul, Minnesota. She received her Ph.D. from the University of Oregon. Before joining the faculty at Macalester, she held a two-year FIPSE post-doctoral position at St. Olaf College in Northfield, Minnesota. She currently serves on the editorial board of the MAA's College Mathematics Journal. This is her first book.
目录
Preface
Introduction: To the Student
1 Metric Spaces, Normed Spaces, Inner Product Spaces
1.1 Basic Definitions and Theorems
1.2 Examples: Sequence Spaces and Function Spaces
1.3 A Discussion About Dimension
Biography: Maurice Frechet
Exercises for Chapter 1
2 The Topology of Metric Spaces
2.1 Open, Closed, and Compact Sets; the Heine-Borel and Ascoli-Arzela Theorems
2.2 Separability
2.3 Completeness: Banach and Hilbert Spaces
Biography: David Hilbert
Biography: Stefan Banach
Exercises for Chapter 2
3 Measure and Integration
3.1 Probability Theory as Motivation
3.2 Lebesgue Measure on Euclidean Space
Biography: Henri Lebesgue
3.3 Measurable and Lebesgue Integrable Functions on Euclidean Space
3.4 The Convergence Theorems
3.5 Comparison of the Lebesgue Integral with the Riemann Integral
3.6 General Measures and the Lebesgue LP-spaces: The
Importance of Lebesgue's Ideas in Functional Analysis
Biography: Frigyes Riesz
Exercises for Chapter 3
4 Fourier Analysis in Hilbert Space
4.1 Orthonormal Sequences
Biography: Joseph Fourier
4.2 Bessel's Inequality, Parseval's Theorem, and the Riesz-Fischer Theorem
4.3 A Return to Classical Fourier Analysis
Exercises for Chapter 4
5 An Introduction to Abstract Linear Operator Theory
5.1 Basic Definitions and Examples
5.2 Boundedness and Operator Norms
5.3 Banach Algebras and Spectra; Compact Operators
5.4 An Introduction to the Invariant Subspace Problem
Biography: Per Enflo
5.5 The Spectral Theorem for Compact Hermitian Operators
Exercises for Chapter 5
6 Further Topics
6.1 The Classical Weierstrass Approximation Theorem and the Generalized Stone-Weierstrass Theorem
Biography: Marshall Stone
6.2 The Baire Category Theorem with an Application to Real Analysis
6.3 Three Classical Theorems from Functional Analysis
6.4The Existence of a Nonmeasurable Set
6.5 Contraction Mappings
6.6 The Function Space C（[a, b]） as a Ring, and its Maximal Ideals
6.7 Hilbert Space Methods in Quantum Mechanics
Biography: John von Nermann
Exercises for Chapter 6
A Complex Numbers
Exercises for Appendix A
B Basic Set Theory
Exercises for Appendix B
References
Index
