Author | Lebedev, L. P. author |
---|---|
Title | Functional Analysis [electronic resource] : Applications in Mechanics and Inverse Problems / by L. P. Lebedev, I. I. Vorovich, G. M. L. Gladwell |
Imprint | Dordrecht : Springer Netherlands, 1996 |
Connect to | http://dx.doi.org/10.1007/978-94-009-0169-8 |
Descript | VIII, 248 p. online resource |
1 Introduction -- 1.1 Real and complex numbers -- 1.2 Theory of functions -- 1.3 Weierstrassโ polynomial approximation theorem -- 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 -- 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 -- 4 Approximation in a Normed Linear Space -- 4.1 Separable spaces -- 4.2 Theory of approximation in a normed linear space -- 4.3 Rieszโs 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 -- 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 -- 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 -- 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 -- 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 Morozovโs discrepancy principle