Through numerous illustrative examples and comments, Applied Functional Analysis, Second Edition demonstrates the rigor of logic and systematic, mathematical thinking. It presents the mathematical foundations that lead to classical results in functional analysis. More specifically, the text prepares students to learn the variational theory of partial differential equations, distributions and Sobolev spaces, and numerical analysis with an emphasis on finite element methods.
While retaining the structure of its best-selling predecessor, this second edition includes revisions of many original examples, along with new examples that often reflect the authors' own vast research experiences and perspectives. This edition also provides many more exercises as well as a solutions manual for qualifying instructors. Each chapter begins with an extensive introduction and concludes with a summary and historical comments that frequently refer to other sources.
New to the Second Edition
Completely revised section on lim sup and lim inf
New discussions of connected sets, probability, Bayesian statistical inference, and the generalized (integral) Minkowski inequality
New sections on elements of multilinear algebra and determinants, the singular value decomposition theorem, the Cauchy principal value, and Hadamard finite part integrals
New example of a Lebesgue non-measurable set
Ideal for a two-semester course, this proven textbook teaches students how to prove theorems and prepares them for further study of more advanced mathematical topics. It helps them succeed in formulating research questions in a mathematically rigorous way.
J. Tinsley Oden is the Director of the Institute for Computational Engineering and Sciences (ICES) and Associate Vice President for Research at The University of Texas at Austin. Dr. Oden is also a member of the US National Academy of Engineering and the recipient of the 2009 SIAM Distinguished Career Award. He holds five honorary doctorates and was knighted Chevalier de l'Ordre des Palmes Academiques by the French government. Leszek F. Demkowicz is the Assistant Director of ICES and a professor in the Department of Aerospace Engineering and Engineering Mechanics at The University of Texas at Austin. Dr. Demkowicz was one of the founding members of the Polish Association for Computational Mechanics (PACM), serving as its first president from 1991 to 1993. His recent research has been summarized in the two volumes Computing with hp-ADAPTIVE FINITE ELEMENTS (CRC Press, 2007 and 2008). For his work on hp methods, he was awarded the 2009 Zienkiewicz Medal by PACM and the 2009 Computational and Applied Sciences Award by the United States Association for Computational Mechanics (USACM).
Preliminaries Elementary Logic and Set Theory Relations Functions Cardinality of Sets Foundations of Abstract Algebra Elementary Topology in Rn Elements of Differential and Integral Calculus Linear Algebra Vector Spaces-The Basic Concepts Linear Transformations Algebraic Duals Euclidean Spaces Lebesgue Measure and Integration Lebesgue Measure Lebesgue Integration Theory Topological and Metric Spaces Elementary Topology Theory of Metric Spaces Banach Spaces Topological Vector Spaces Hahn-Banach Extension Theorem Bounded (Continuous) Linear Operators on Normed Spaces Closed Operators Topological Duals. Weak Compactness Closed Range Theorem. Solvability of Linear Equations Hilbert Spaces Basic Theory Duality in Hilbert Spaces Elements of Spectral Theory References