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  n
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 n ................ 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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