Applied Functional Analysis

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).

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