Functional Analysis

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Courier Dover Publications, 1990 - Mathematics - 504 pages
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Classic exposition of modern theories of differentiation and integration and the principal problems and methods of handling integral equations and linear functionals and transformations. Topics include Lebesque and Stieltjes integrals, Hilbert and Banach spaces, self-adjunct transformations, spectral theories for linear transformations of general type, more. Translated from 2nd French edition by Leo F. Boron. 1955 edition. Bibliography.
  

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Contents

DIFFERENTIATION
3
Some Immediate Consequences of Lebesgues Theorem
11
Interval Functions
19
THE LEBESGUE INTEGRAL Definition and Fundamental Properties
29
The Integral for Summable Functions
31
TermbyTerm Integration of an Increasing Sequence Beppo Levis Theorem
33
TermbyTerm Integration of a Majorited Sequence Le besgues Theorem
36
Theorems Affirming the Integrability of a Limit Function
38
Uniqueness of the Generating Function
111
Extension of a Linear Functional
112
The Approximation Theorem Moment Problems
115
Integration by Parts The Second Theorem of the Mean
118
Sequences of Functionals
119
Generalitation of the Stieltjes Integral
122
Reduction of the LebesgueStieltjes Integral to That of Lebesgue
124
Relations Between Two LebesgueStieltjes Integrals
126

The Schwart Holder and Minkowski Inequalities
40
The Derivative Over a Net of a Nonnegative Additive
41
Measurable Sets and Measurable Functions
43
Indefinite Integrals Absolutely Continuous Functions
47
Example of a Monotonic Continuous Function Whose Derivative Is Zero Almost Everywhere
48
Absolutely Continuous Functions Canonical Decomposition of Monotonic Functions
50
Integration by Parts and Integration by Substitution
54
The Integral as a Set Function
56
The Space L1 and its Linear Functionals L Spaces
57
Weak Convergence
60
Linear Functionals
61
Sequence of Linear Functionals a Theorem of Osgood
63
Separability of La The Theorem of Choice
64
Orthonormal Systems
66
Subspaces of L The Decomposition Theorem
71
Another Proof of the Theorem of Choice Extension of Functionals
72
The Space L and Its Linear Functionals
73
A Theorem on Mean Convergence
78
A Theorem of Banach and Saks
80
Functions of Several Variables
81
Successive Integrations Fubinis Theorem
83
Rectangle Function Parallel Displacement of the Net
84
Rectangle Functions of Bounded Variation Conjugate Nets
87
Additive Set Functions Sets Measurable B
89
Other Definitions of the Lebesgue Integral
92
Functions Measurable L and the Integral L
94
Other Definitions Egoroffs Theorem
96
Elementary Proof of the Theorems of Arxela and Osgood
100
The Lebesgue Integral Considered as the Inverse Operation of Differentiation
103
THE STIELTJES INTEGRAL AND ITS GENERALIZATIONS Linear Functionals on the Space of Continuous Functions
105
Linear Functionals on the Space C
106
Functions of Several Variables Direct Definition
128
Definition by Means of the Principle of Transition
130
The Daniel Integral
132
Functionals of Variable Sign
134
The Derivative of One Linear Functional With Respect to Another
137
INTEGRAL EQUATIONS
143
The Fredholm Alternative
161
Fredholm Determinants
172
Applications to Potential Theory
190
Hilbert Space
195
Banach Spaces
210
BOUNDED SYMMETRIC UNITARY
261
COMPLETELY CONTINUOUS SYMMETRIC
227
Solution of the Functional Equation L4 g
235
Transformations with Symmetric Kernel
242
Unitary and Normal Transformations
280
Unitary Transformations of the Space L1
291
SelfAdjoint Transformations Spectral Decomposition
308
Extensions of Symmetric Transformations
325
SELFADJOINT TRANSFORMATIONS
341
Characteristic Properties of Functions of a SelfAdjoint
351
The Spectrum of a SelfAdjoint Transformation and Its Pertur
360
Unitary Transformations
380
NonUnitary Transformations
393
Ergodic Theorems
406
SPECTRAL THEORIES FOR LINEAR
415
Von Neumanns Theory of Spectral Sets
435
Bibliography
447
Appendix
457
Index
493
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