Functional analysis. Applications in mechanics and inverse problems
This textbook is intended for engineers and applied mechanicists who wish to understand the functional analysis used in contemporary research in their fields. The book does not tell about functional analysis in its most abstract form, it is rather a tour through those parts of the analysis needed for applications. After a short introductory chapter 1 (real and complex numbers, theory of functions, Weierstrass' polynomial approximation theorem), chapter 2 introduces metric spaces, normed linear space, inner product spaces and the concepts of open and closed sets and completeness. The concept of a compact set, which was introduced in chapter 1 for real numbers, is not introduced until chapter 4, and not discussed fully until chapter 6. Chapter 3 stands somewhat apart from the others; it illustrates how the idea of imbedding, appearing in Sobolev's theory, arises in continuum analysis. Among other things, here are discussed rods, Euler-Bernoulli beams, membranes, plates in bending and linear elasticity). From chapter 2 the reader may pass directly to chapter 4 which considers the important problem of approximation, introduces Riesz's representation theorem for linear functionals, and the concept of weak convergence in a Hilbert space. The presentation of the concepts of weak convergence, and of the adjoint operator in chapter 5, are limited to inner product spaces. The theory of linear operators is discussed, but not covered (!), in chapters 5 and 7. The emphasis here is on the parts of the theory related to compact linear operators and self-adjoint linear operators. The final chapter 8 is devoted to inverse problems. It presents well-posed and ill-posed problems, operator equations, singular value decomposition, regularization and Morozov's discrepancy principle. It is the authors' fervent wish that readers will find the book enjoyable and instructive, and allow them to use functional analysis methods in their own research, or to use the book as a jumping board to more advanced and/or abstract texts.
Weierstrass' polynomial approximation theorem
scientific article; zbMATH DE number 863427