Verfügbarkeit: Beginning functional analysis

Beginning functional analysis:

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Bibliographische Detailangaben
1. Verfasser:	Saxe, Karen (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York ; Berlin [u.a.] Springer 2002
Schriftenreihe:	Undergraduate texts in mathematics
Schlagworte:	Funktionalanalysis - Lehrbuch Functional analysis Funktionalanalysis
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XI, 197 S. Ill., graph. Darst.
ISBN:	0387952241

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MARC


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adam_text	CONTENTS PREFACE V INTRODUCTION: TO THE STUDENT 1 1 METRIC SPACES, NORMED SPACES, INNER PRODUCT SPACES 3 1.1 BASIC DEFINITIONS AND THEOREMS . . . . . . . . . . . . . . . . . 4 1.2 EXAMPLES: SEQUENCE SPACES AND FUNCTION SPACES . . . . . . . 8 1.3 A DISCUSSION ABOUT DIMENSION . . . . . . . . . . . . . . . . . 11 BIOGRAPHY: MAURICE FR´ ECHET . . . . . . . . . . . . . . . . . . . . . . 11 EXERCISES FOR CHAPTER 1 . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 THE TOPOLOGY OF METRIC SPACES 16 2.1 OPEN, CLOSED, AND COMPACT SETS; THE HEINEBOREL AND ASCOLIARZEL` A THEOREMS . . . . . . . . . . . . . . . . . . . . 16 2.2 SEPARABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 COMPLETENESS: BANACH AND HILBERT SPACES . . . . . . . . . . . 23 BIOGRAPHY: DAVID HILBERT . . . . . . . . . . . . . . . . . . . . . . . . 24 BIOGRAPHY: STEFAN BANACH . . . . . . . . . . . . . . . . . . . . . . . 28 EXERCISES FOR CHAPTER 2 . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 MEASURE AND INTEGRATION 32 3.1 PROBABILITY THEORY AS MOTIVATION . . . . . . . . . . . . . . . . 33 3.2 LEBESGUE MEASURE ON EUCLIDEAN SPACE . . . . . . . . . . . . . 35 BIOGRAPHY: HENRI LEBESGUE . . . . . . . . . . . . . . . . . . . . . . . 42 X CONTENTS 3.3 MEASURABLE AND LEBESGUE INTEGRABLE FUNCTIONS ON EUCLIDEAN SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 THE CONVERGENCE THEOREMS . . . . . . . . . . . . . . . . . . . 50 3.5 COMPARISON OF THE LEBESGUE INTEGRAL WITH THE RIEMANN INTEGRAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.6 GENERAL MEASURES AND THE LEBESGUE L P -SPACES: THE IMPORTANCE OF LEBESGUES IDEAS IN FUNCTIONAL ANALYSIS . . . . 57 BIOGRAPHY: FRIGYES RIESZ . . . . . . . . . . . . . . . . . . . . . . . . 68 EXERCISES FOR CHAPTER 3 . . . . . . . . . . . . . . . . . . . . . . . . . 69 4 FOURIER ANALYSIS IN HILBERT SPACE 74 4.1 ORTHONORMAL SEQUENCES . . . . . . . . . . . . . . . . . . . . . 74 BIOGRAPHY: JOSEPH FOURIER . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 BESSELS INEQUALITY, PARSEVALS THEOREM, AND THE RIESZFISCHER THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 A RETURN TO CLASSICAL FOURIER ANALYSIS . . . . . . . . . . . . . 85 EXERCISES FOR CHAPTER 4 . . . . . . . . . . . . . . . . . . . . . . . . . 88 5 AN INTRODUCTION TO ABSTRACT LINEAR OPERATOR THEORY 92 5.1 BASIC DEFINITIONS AND EXAMPLES . . . . . . . . . . . . . . . . . 92 5.2 BOUNDEDNESS AND OPERATOR NORMS . . . . . . . . . . . . . . . 95 5.3 BANACH ALGEBRAS AND SPECTRA; COMPACT OPERATORS . . . . . . . 99 5.4 AN INTRODUCTION TO THE INVARIANT SUBSPACE PROBLEM . . . . . . . 112 BIOGRAPHY: PER ENFLO . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.5 THE SPECTRAL THEOREM FOR COMPACT HERMITIAN OPERATORS . . . 124 EXERCISES FOR CHAPTER 5 . . . . . . . . . . . . . . . . . . . . . . . . . 129 6 FURTHER TOPICS 135 6.1 THE CLASSICAL WEIERSTRASS APPROXIMATION THEOREM AND THE GENERALIZED STONE*WEIERSTRASS THEOREM . . . . . . . . . . . . 136 BIOGRAPHY: MARSHALL STONE . . . . . . . . . . . . . . . . . . . . . . . 145 6.2 THE BAIRE CATEGORY THEOREM WITH AN APPLICATION TO REAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.3 THREE CLASSICAL THEOREMS FROM FUNCTIONAL ANALYSIS . . . . . . 151 6.4 THE EXISTENCE OF A NONMEASURABLE SET . . . . . . . . . . . . . 158 6.5 CONTRACTION MAPPINGS . . . . . . . . . . . . . . . . . . . . . . 160 6.6 THE FUNCTION SPACE C ([ A, B ]) AS A RING, AND ITS MAXIMAL IDEALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.7 HILBERT SPACE METHODS IN QUANTUM MECHANICS . . . . . . . . . 167 BIOGRAPHY: JOHN VON NEUMANN . . . . . . . . . . . . . . . . . . . . . 174 EXERCISES FOR CHAPTER 6 . . . . . . . . . . . . . . . . . . . . . . . . . 176 A COMPLEX NUMBERS 181 EXERCISES FOR APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . 183 CONTENTS XI B BASIC SET THEORY 185 EXERCISES FOR APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . 186 REFERENCES 187 INDEX 193
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title	Beginning functional analysis
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title_fullStr	Beginning functional analysis Karen Saxe
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