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基本説明
Textbook.
Full Description
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.
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
