Applied Functional Analysis, Second Edition

Applied Functional Analysis, Second Edition
  • e-Book: 596 pages
  • Also available in Hardback and e-Book
  • Published: September 2011
  • ISBN: 978-1-4398827-0-2
  • Publisher: Chapman and Hall/CRC

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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.

Table of Contents

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

Reviews

The textbook is designed to drive a crash course for beginning graduate students majoring in something besides mathematics, introducing mathematical foundations that lead to classical results in functional analysis. More specifically, Oden and Demkowicz want to prepare students to learn the variational theory of partial differential equations, distributions, and Sobolev spaces and numerical analysis with an emphasis on finite element methods. The 1996 first edition has been used in a rather intensive two-semester course.

—Book News, June 2010

Author/Editor Biography

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 Académiques 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).