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Table of Contents |
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1 Introduction.................................................................................. 1.1 Real and complex numbers....................................................... 1.2 Theory of functions................................................................. 1.3 Weierstrass' polynomial approximation theorem.......................... |
1 1 9 14 |
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2 Introduction to Metric Spaces.................................................... 2.1 Preliminaries............................................................................ 2.2 Sets in a metric space............................................................... 2.3 Some metric spaces of functions............................................... 2.4 Convergence in a metric space.................................................. 2.5 Complete metric spaces............................................................ 2.6 The completion theorem............................................................ 2.7 An introduction to operators................................................. 2.8 Normed linear spaces................................................................ 2.9 An introduction to linear operators...................................... 2.10 Some inequalities..................................................................... 2.11 Lebesgue spaces...................................................................... 2.12 Inner product spaces........................................................... |
19 19 25 27 29 30 32 35 40 45 48 51 58 |
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3 Energy Spaces and Generalized Solutions ........................................ 3.1 The rod............................................................................... 3.2 The Euler-Bernoulli beam................................................... 3.3 The membrane.................................................................... . 3.4 The plate in bending.......................................................... 3.5 Linear elasticity..................................................................... 3.6 Sobolev spaces ................................................................. 3.7 Some imbedding theorems..................................................... |
65 65 74 78 83 85 88 90 |
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4 Approximation in a Normed Linear Space ......................................... 4.1 Separable spaces .................................................................. 4.2 Theory of approximation in a normed linear space...................... 4.3 Riesz's representation theorem.................................................. 4.4 Existence of energy solutions of some mechanics problems ........ 4.5 Bases and complete systems..................................................... 4.6 Weak convergence in a Hilbert space.................................. 4.7 Introduction to the concept of a compact set..................... 4.8 Ritz approximation in a Hilbert space................................. 4.9 Generalized solutions of evolution problems...................... |
99 99 103 106 110 113 120 126 128 132 |
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5 Elements of the Theory of Linear Operators ........................... 5.1 Spaces of linear operators ............................................... 5.2 The Banach-Steinhaus theorem........................................... 5.3 The inverse operator............................................................ 5.4 Closed operators .............................................................. 5.5 The adjoint operator............................................................ 5.6 Examples of adjoint operators............................................. |
141 141 144 147 152 157 162 |
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6 Compactness and Its Consequences ......................................... 6.1 Sequentially compact = compact........................................ 6.2 Criteria for compactness..................................................... 6.3 The Arzela-Ascoli theorem................................................. 6.4 Applications of the Arzela-Ascoli theorem......................... 6.5 Compact linear operators in normed linear spaces............ 6.6 Compact linear operators between Hilbert spaces .......... |
167 167 171 174 178 183 189 |
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7 Spectral Theory of Linear Operators ....................................... 7.1 The spectrum of a linear operator...................................... 7.2 The resolvent set of a closed linear operator..................... 7.3 The spectrum of a compact linear operator in a Hilbert space ............................................................... 7.4 The analytic nature of the resolvent of a compact linear operator................................................... 7.5 Self-adjoint operators in a Hilbert space ........................ |
195 195 199
201
208 211 |
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8 Applications to Inverse Problems ....................................... 8.1 Well-posed and ill-posed problems.................................... 8.2 The operator equation........................................................ 8.3 Singular value decomposition ....................................... 8.4 Regularization.................................................................... 8.5 Morozov's discrepancy principle...................................... |
219 219 220 226 229 234 |
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Index ............................................................................................ |
241 |