This work is for readers who want to use functional analysis to justify approximate methods in mechanics and inverse problems. It provides such researchers with the tools they need without having to assimilate or skip through concepts they do not need. The essence of functional analysis is abstraction: from the everyday ideas of 3-dimensional space and distance, one abstracts the concepts of metric space and metric. The properties of this metric are laid down as axioms on which all subsequent arguments are based. The vocabulary of functional analysis consists largely of terms which originally appeared either in geometry or in connection with the real line: set, closed, open, bounded, compact, inner-product, etc.; in functional analysis they are defined abstractly. The book starts from scratch with a chapter on real numbers and functions. Chapter 2 introduces metric spaces, including the concept of a complete space and Banach's contraction mapping theorem; normed linear spaces, and inner product spaces. An excursion into some boundary value problems in mechanics leads up to the concept of a generalized solution, and to Sobolev space.
A study of approximation in Hilbert space leads to Riesz's representation theorem. An introduction to linear operators is followed by a chapter on the essential, but often misunderstood concept of a compact set. En route the mysteries of weakly closed, weakly convergent, sequential compactness, compact operator and singular value decomposition are revealed. The final chapter shows how the language of functional analysis is ideally suited to elucidate and justify the regularization methods for the ill-posed inverse problems exemplified by Fredholm integral equations of the first kind.
Publisher: Kluwer Academic Publishers