Functional Analysis for the Applied Mathematician

This self-contained volume provides a rigorous introduction to functional analysis and its applications. It is accessible to graduate and advanced undergraduate students with a solid background in undergraduate mathematics and an appreciation of mathematical rigor.

Functional Analysis for the Applied Mathematician is a self-contained volume providing a rigorous introduction to functional analysis and its applications. Students from mathematics, science, engineering, and certain social science and interdisciplinary programs will benefit from the material. It is accessible to graduate and advanced undergraduate students with a solid background in undergraduate mathematics and an appreciation of mathematical rigor. Students are called upon to actively engage with the material, to the point of proving some of the basic results or their straightforward generalizations, both within the text and within the generous set of exercises.

Features:

• Replete with exercises and examples
• Suitable for graduate students and advanced undergraduates
• Develops the basics of functional analysis, exploring the interplay between algebraic linear space theory and topology
• Presents a variety of applications, often dealing with partial differential equations and their numerical approximation

• Doubles as a reference book with an extensive index listing the concepts and results

  Autorentext

  Todd Arbogast, Professor of Mathematics, born December 9, 1957. He earned his Ph.D. from the University of Chicago in 1987 under the direction of Professor Jim Douglas, Jr. He has held faculty positions at Purdue University, Rice University, and the University of Texas at Austin, where he is a core member of the Oden Institute for Computational Engineering and Sciences. He is a Fellow of both the American Mathematical Society and the Society for Industrial and Applied Mathematics. His research contributes to the development and analysis of numerical algorithms for the approximation of partial differential systems, homogenization and multiscale modeling, and scientific computation, as well as applications of the same to the modeling and simulation of multiphase flow and transport through geologic porous media.

Jerry L. Bona, Professor of Mathematics, Statistics & Computer Science, was born on February 5, 1945. He earned his Ph.D. in 1971 from Harvard University under the supervision of Professor Garrett Birkhoff. His early work in the Fluid Mechanics Research Institute at the University of Essex with Professors Brooke Benjamin and J. J. Mahony resulted in a model equation for long waves in non-linear dispersive systems, known as the Benjamin-Bona-Mahony equation. He has held faculty positions at the University of Chicago, the Pennsylvania State University, the University of Texas at Austin, and the University of Illinois at Chicago. He is a fellow of the American Mathematical Society, the Society for Industrial and Applied Mathematics, the Centre de Recherche Mathématiques of the Université de Montréal, and the American Association for the Advancement of Science. His research is in fluid mechanics, oceanography, coastal engineering, mathematical aspects of biology, mathematical economics, and the associated theory of partial differential equations, computational mathematics, and numerical analysis.

Inhalt

1. Preliminaries. 2. Normed Linear Spaces and Banach Spaces. 3. Hilbert Spaces. 4. Spectral Theory and Compact Operators. 5. Distributions. 6. The Fourier Transform. 7. Sobolev Spaces. 8. Boundary Value Problems. 9. Differential Calculus in Banach Spaces. 10. The Calculus of Variations.