Integral and Functional Analysis

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Nova Publishers, 2008 - Mathematics - 287 pages
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This book is based on two closely-related courses. The first of these courses is Integration and Metric Spaces, and the second being Functional Analysis. Though the contents of Functional Analysis have been used for both an undergraduate course and an introductory graduate course, this text is designed primarily for undergraduate students. The prerequisites of this book are deliberately modest, and it is assumed that the students have some familiarity with Introductory Calculus and Linear Algebra plus the basic (direct, indirect) proof methods.
 

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Contents

Preliminaries
1
12 Reals Some Basic Theorems and Sequence Limits
8
Problems
15
Riemann Integrals
17
22 Algebraic Operations and the Darboux Criterion
23
23 Fundamental Theorem of Calculus
28
24 Improper Integrals
35
Problems
42
62 Continuous Mappings on Compact or Connected Spaces
133
63 Sequences of Mappings
135
64 Contractions
139
65 Equivalence of Metric Spaces
144
Problems
146
Normed Linear Spaces
149
72 Finite Dimensional Spaces
156
73 Bounded Linear Operators
160

RiemannStieltjes Integrals
45
32 Definition and Basic Properties
48
33 Nonexistence and Existence for Integrals
51
34 Evaluation of Integrals
59
35 Improper Cases
62
Problems
66
LebesgueRadonStieltjes Integrals
69
41 Foundational Material
70
42 Essential Properties
76
43 Convergence Theorems
80
44 Extension via Measurability
86
45 Double and Iterated Integrals with Applications
91
Problems
102
Metric Spaces
109
52 Completeness
115
53 Compactness Density and Separability
120
Problems
127
Continuous Maps
129
74 Linear Functionals via HahnBanach Extension
163
Problems
169
Banach Spaces via Operators and Functionals
173
82 Uniform Boundedness Open Mapping and Closed Graph
178
83 Dual Banach Spaces by Examples
182
84 Weak and Weakstar Topologies
192
85 Compact and Dual Operators
197
Problems
200
Hilbert Spaces and Their Operators
205
92 Orthogonality Orthogonal Complement and Duality
208
93 Orthonormal Sets and Bases
211
94 Five Special Bounded Operators
217
95 Compact Operators via the Spectrum
225
Problems
233
Hints and Solutions
239
Bibliography
281
Index
285
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