Elements of the Theory of Functions and Functional Analysis, Band 1

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Based on the authors' courses and lectures, this two-part advanced-level text is now available in a single volume. Topics include metric and normed spaces, continuous curves in metric spaces, measure theory, Lebesque intervals, Hilbert space, and more. Each section contains exercises. Lists of symbols, definitions, and theorems. 1957 edition.
 

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Inhalt

The Concept of Set Operations on Sets
1
Finite and Infinite Sets Denumerability
3
Equivalence of Sets
6
The Nondenumerability of the Set of Real Numbers
8
The Concept of Cardinal Number
9
Partition into Classes
11
Mappings of Sets General Concept of Function
13
CHAPTER II
16
Convex Sets in Normed Linear Spaces
74
Linear Functionals
77
The Conjugate Space
81
Extension of Linear Functionals
86
The Second Conjugate Space
88
Weak Convergence
90
Weak Convergence of Linear Functionals
92
Linear Operators
95

Convergence of Sequences Limit Points
23
Open and Closed Sets
26
Open and Closed Sets on the Real Line
31
Continuous Mappings Homeomorphism Isometry
33
Complete Metric Spaces
36
The Principle of Contraction Mappings and its Applications
43
Applications of the Principle of Contraction Mappings in Analysis
46
Compact Sets in Metric Spaces
51
Arzelàs Theorem and its Applications
53
Compacta
57
Real Functions in Metric Spaces
62
Continuous Curves in Metric Spaces
66
CHAPTER III
71
Spectrum of an Operator Resolvents
110
Linear Operator Equations Fredholms Theorems
117
List of DEFINITIONs
123
MEASURE THEORY
Collections of sets
15
Extension of Jordan measure
25
39
31
MEASURABLE FUNCTIONS
38
CHAPTER VII
48
Passage to the limit under the Lebesgue integral
56
Products of sets and measures
65
Fubinis theorem
72
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