Bryan Rynne – författare

This book provides an introduction to the ideas and methods of linear func­ tional analysis at a level appropriate to the final year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the the­ ory of metric spaces). Part of the development of functional analysis can be traced to attempts to find a suitable framework in which to discuss differential and integral equa­ tions. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions defined on some set. In general, such a vector space is infinite-dimensional. This leads to difficulties in that, although many of the elementary properties of finite-dimensional vector spaces hold in infinite­ dimensional vector spaces, many others do not. For example, in general infinite­ dimensional vector spaces there is no framework in which to make sense of an­ alytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now briefly outline the contents of the book.

This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the ?nal year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the t- ory of metric spaces). Part of the development of functional analysis can be traced to attempts to ?nd a suitable framework in which to discuss di?erential and integral equations. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions de?ned on some set. In general, such a v- tor space is in?nite-dimensional. This leads to di?culties in that, although many of the elementary properties of ?nite-dimensional vector spaces hold in in?nite-dimensional vector spaces, many others do not. For example, in general in?nite-dimensionalvectorspacesthereisnoframeworkinwhichtomakesense of analytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now brie?y outline the contents of the book.

This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the ?nal year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the t- ory of metric spaces). Part of the development of functional analysis can be traced to attempts to ?nd a suitable framework in which to discuss di?erential and integral equations. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions de?ned on some set. In general, such a v- tor space is in?nite-dimensional. This leads to di?culties in that, although many of the elementary properties of ?nite-dimensional vector spaces hold in in?nite-dimensional vector spaces, many others do not. For example, in general in?nite-dimensionalvectorspacesthereisnoframeworkinwhichtomakesense of analytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now brie?y outline the contents of the book.

This volume consists ofsurvey articles on current topics in computational wave prop­ agation and inverse problems, written by leading experts in their respective fields. The idea to compile such a volume arose in conjunction with the LMS Durham Symposium on Computational Methods for Wave Propagation in Direct Scattering held at the University of Durham from 15th-25th July 2002, which we jointly or­ ganised. The meeting, attended by 70 participants from the UK and overseas, was structured around a number of short, three lecture, survey courses on a range of top­ ics on computational wave propagation and inverse problems beginning at the level of a graduate student. We were delighted to secure the participation of distinguished international researchers to present these lectures. We felt that it would be valuable to record this material for the benefit of a wider audience, and the idea was hatched that the individual lecturers should be invited to contribute a survey article. Fortunately, many of the speakers not only agreed to undertake this arduous task, but produced what we hope you will agree are the high quality contributions found in this volume. Finally, it is a pleasure to thank the Engineering and Physical Sciences Research Council of Great Britain and the London Mathematical Society for providing the generous support that allowed the meeting to take place. Mark Ainsworth Glasgow, 2003 Penny Davies Dugald Duncan Paul Martin Bryan Rynne Contents New Results on Absorbing Layers and Radiation Boundary Conditions Thomas Hagstrom .