Oden J.T. Applied functional analysis (Boca Raton; London, 2010). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаOden J.T. Applied functional analysis. - 2nd ed. - Boca Raton; London: CRC Press/Taylor & Francis, 2010. - xvii, 578 p.: ill. - Ref.: p.569. - Ind.: p.570-578. - ISBN 978-1-4200-9195-3
 

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Оглавление / Contents
 
1  Preliminaries ................................................ 1
   Elementary  Logic  and  Set  Theory
   1.1  Sets and Preliminary Notations, Number Sets ............. 1
   1.2  Level One Logic ......................................... 3
   1.3  Algebra of Sets ......................................... 9
   1.4  Level Two Logic ........................................ 16
   1.5  Infinite Unions and Intersections ...................... 18
   Relations
   1.6  Cartesian Products, Relations .......................... 21
   1.7  Partial Orderings ...................................... 26
   1.8  Equivalence Relations, Equivalence Classes,
        Partitions ............................................. 31
   Functions
   1.9  Fundamental Definitions ................................ 36
   1.10 Compositions, Inverse Functions ........................ 43
   Cardinality of Sets
   1.11 Fundamental Notions .................................... 52
   1.12 Ordering of Cardinal Numbers ........................... 54
   Foundations of Abstract Algebra
   1.13 Operations, Abstract Systems, Isomorphisms ............. 58
   1.14 Examples of Abstract Systems ........................... 63
   Elementary Topology in fig.1n
   1.15 The Real Number System ................................. 73
   1.16 Open and Closed Sets ................................... 78
   1.17 Sequences .............................................. 85
   1.18 Limits and Continuity .................................. 92
   Elements of Differential and Integral Calculus
   1.19 Derivatives and Integrals of Functions of One
        Variable .............. ................................ 97
   1.20 Multidimensional  Calculus ............................ 104
2  Linear Algebra ............................................. 111
   Vector Spaces-The Basic Concepts
   2.1  Concept of a Vector Space ............................. 111
   2.2  Subspaces ............................................. 118
   2.3  Equivalence Relations and Quotient Spaces ............. 123
   2.4  Linear Dependence and Independence, Hamel Basis,
        Dimension ............................................. 129
   Linear Transformations
   2.5  Linear Transformations-The Fundamental Facts .......... 139
   2.6  Isomorphic Vector Spaces .............................. 146
   2.7  More About Linear Transformations ..................... 150
   2.8  Linear Transformations and Matrices ................... 156
   2.9  Solvability of Linear Equations ....................... 159
   Algebraic Duals
   2.10 The Algebraic Dual Space, Dual Basis .................. 162
   2.11 Transpose of a Linear Transformation .................. 170
   2.12 Tensor Products, Covariant and Contravariant
        Tensors ............................................... 176
   2.13 Elements of Multilinear Algebra ....................... 181
   Euclidean Spaces
   2.14 Scalar (Inner) Product, Representation Theorem in
        Finite-Dimensional Spaces ............................. 188
   2.15 Basis and Cobasis, Adjoint of a Transformation,
        Contra- and Covariant Components of Ten-sors .......... 192
3  Lebesgue Measure and Integration ........................... 201
   Lebesgue  Measure
   3.1  Elementary Abstract Measure Theory .................... 201
   3.2  Construction of Lebesgue Measure in fig.1n ................ 210
   3.3  The Fundamental Characterization of Lebesgue
        Measure ............................................... 221
   Lebesgue  Integration  Theory
   3.4  Measurable and Borel Functions ........................ 230
   3.5  Lebesgue Integral of Nonnegative Functions ............ 233
   3.6  Fubini's Theorem for Nonnegative Functions ............ 238
   3.7  Lebesgue Integral of Arbitrary Functions .............. 245
   3.8  Lebesgue Approximation Sums, Riemann Integrals ........ 254
   IP Spaces
   3.9  Holder and Minkowski Inequalities ..................... 260
4  Topological and Metric Spaces .............................. 269
   Elementary Topology
   4.1  Topological Structure-Basic Notions ................... 269
   4.2  Topological Subspaces and Product Topologies .......... 287
   4.3  Continuity and Compactness ............................ 291
   4.4  Sequences ............................................. 301
   4.5  Topological Equivalence. Homeomorphism ................ 306
   Theory of Metric Spaces
   4.6  Metric and Normed Spaces, Examples .................... 308
   4.7  Topological Properties  of  Metric  Spaces ............ 316
   4.8  Completeness and  Completion  of  Metric  Spaces ...... 321
   4.9  Compactness in Metric Spaces .......................... 333
   4.10 Contraction Mappings and Fixed Points ................. 346
5  Banach  Spaces ............................................. 355
   Topological Vector Spaces
   5.1  Topological Vector Spaces-An Introduction ............. 355
   5.2  Locally Convex Topological Vector Spaces .............. 357
   5.3  Space  of  Test  Functions ............................ 364
   Hahn-Banach Extension Theorem
   5.4  The Hahn-Banach Theorem ............................... 368
   5.5  Extensions  and  Corollaries .......................... 371
   Bounded (Continuous) Linear Operators on Normed Spaces
   5.6  Fundamental Properties of Linear Bounded Operators .... 374
   5.7  The Space of Continuous Linear Operators .............. 382
   5.8  Uniform Boundedness and Banach-Steinhaus Theorems ..... 387
   5.9  The Open Mapping Theorem .............................. 389
   Closed Operators
   5.10 Closed Operators, Closed Graph Theorem ................ 392
   5.11 Example of a Closed Operator .......................... 398
   Topological Duals. Weak Compactness
   5.12 Examples of Dual Spaces, Representation Theorem for
        Topological Duals of Lp Spaces ........................ 401
   5.13 Bidual, Reflexive Spaces .............................. 412
   5.14 Weak Topologies, Weak Sequential Compactness .......... 417
   5.15 Compact (Completely Continuous) Operators ............. 425
   Closed Range Theorem. Solvability of Linear Equations
   5.16 Topological Transpose Operators, Orthogonal
        Complements ........................................... 430
   5.17 Solvability of Linear Equations in Banach Spaces,
        The Closed Range Theorem .............................. 434
   5.18 Generalization for Closed Operators ................... 439
   5.19 Examples .............................................. 443
   5.20 Equations with Completely Continuous Kernels.
        Fredholm  Alternative ................................. 449
6  Hilbert Spaces ............................................. 463
   Basic Theory
   6.1  Inner Product and Hilbert Spaces ...................... 463
   6.2  Orthogonality and Orthogonal Projections .............. 480
   6.3  Orthonormal Bases and Fourier Series .................. 487
   Duality in Hilbert Spaces
   6.4  Riesz Representation Theorem .......................... 498
   6.5  The Adjoint of a Linear Operator ...................... 506
   6.6  Variational Boundary-Value Problems ................... 516
   6.7  Generalized Green's Formulas for Operators on
        Hilbert Spaces ........................................ 530
   Elements of Spectral Theory
   6.8  Resolvent Set and Spectrum ............................ 540
   6.9  Spectra of Continuous Operators. Fundamental
        Properties ............................................ 545
   6.10 Spectral Theory for Compact Operators ................. 550
   6.11 Spectral Theory for Self-Adjoint Operators ............ 560
7  References ................................................. 569
Index ......................................................... 570


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