Functional analysis. Applications in mechanics and inverse problems
type: Item
instance of: scholarly article
title: Functional analysis. Applications in mechanics and inverse problems
zbMATH Open document ID: 0843.73001
author: Leonid Lebedev
author: I. I. Vorovich
author: Graham M. L. Gladwell
published in: Solid Mechanics and Its Applications
publication date: 1996-04-08T00:00:00Z
review text: 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 mor…
Mathematics Subject Classification ID: 74-01
Mathematics Subject Classification ID: 74Kxx
Mathematics Subject Classification ID: 46N60
Mathematics Subject Classification ID: 47N60
zbMATH DE Number: 863427
zbMATH Keywords: real and complex numbers
zbMATH Keywords: theory of functions
zbMATH Keywords: Weierstrass' polynomial approximation theorem
zbMATH Keywords: metric spaces
zbMATH Keywords: normed linear space
zbMATH Keywords: compact set
zbMATH Keywords: imbedding
zbMATH Keywords: rods
zbMATH Keywords: Euler-Bernoulli beams
zbMATH Keywords: membranes
zbMATH Keywords: plates in bending
zbMATH Keywords: linear elasticity
zbMATH Keywords: Riesz's representation theorem
zbMATH Keywords: weak convergence
zbMATH Keywords: Hilbert space
zbMATH Keywords: adjoint operator
zbMATH Keywords: compact linear operators
zbMATH Keywords: self-adjoint linear operators
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 863427