Functional Analysis: Applications in Mechanics and Inverse Problems(41 Solid Mechanics and Its Applications)

This book started its life as a series of lectures given by the second author from the 1970's onwards to students in their third and fourth years in the Department of Mathematics at the Rostov State University. For these lectures there was also an audience of engineers and applied mechanicists who wished to understand the functional analysis used in contemporary research in their fields. These people were not so much interested in functional analysis itself as in its applications; they did not want to be told about functional analysis in its most abstract form, but wanted a guided tour through those parts of the analysis needed for their applications. The lecture notes evolved over the years as the first author started to make more formal typewritten versions incorporating new material. About 1990 the first author prepared an English version and submitted it to Kluwer Academic Publishers for inclusion in the series Solid Mechanics and its Applications. At that stage the notes were divided into three long chapters covering linear and nonlinear analysis. As Series Editor, the third author started to edit them.

Table of Contents:
1 Introduction.- 1.1 Real and complex numbers.- 1.2 Theory of functions.- 1.3 Weierstrass’ 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 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.- 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 resolventset 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 Morozov’s discrepancy principle.