Linear operator theory in engineering and science.
This book is an elementary introduction to classical functional analysis in a form suitable for engineers, scientists, and applied mathematicians. The main part of this book is related to motivation of basic definitions and constructions of functional analysis, it consists of many illustrative examples. The style of the book is vivid and easily understandable; in the book there are presented numerous exercises, which, beyond any possible doubt, have greatly advanced the understanding of material presented in the book. The book consists of 7 chapters and 5 appendices. Chapter 1, ``Introduction'', really describes the authors' approach to the problems discussed in this book; here we emphasize only that the authors begin from the engineer systems (``black boxes'') and explain the rôle of the axiomatic method in what follows. Chapter 2, ``Set-Theoretic Structure'', deals with basic notions, like sets and operations with them, functions, systems and so on. Chapter 3, ``Topological Structure'', is a nice account of the theory of metric spaces including the contraction mapping theorem. Chapter 4, ``Algebraic Structure'', presents the theory of linear spaces and operators between them. Chapter 5, ``Combined Topological and Algebraic Structure'', deals with the elementary theory of Banach and Hilbert spaces, theory of special classes of linear operators (normal and self-adjoint, compact and so on). Chapter 6, ``Analysis of Linear Operators (Compact Case)'', is devoted to standard problems related to linear operators: the spectrum and resolvent, spectral theorem for compact self-adjoint operators, functions of operators, some results about nonnormal operators. The last chapter 7, ``Analysis of Unbounded Operators'', presents elements of the theory of symmetric operators and its applications to ordinary and partial differential equations and quantum mechanics, including the Heisenberg Uncertainty Principle and the theory of harmonic oscillators. Appendices are devoted to Hölder, Schwartz, and Minkowski inequalities, Cardinality and Zorn's lemma, elements of integration and measure theory, and probabilistic spaces and stochastic processes. The book can be used by students with various level of preparation with different interests, as well as a textbook for a senior-level courses.
elementary theory of Banach and Hilbert spaces
theorem for compact self-adjoint operators
ordinary and partial differential equations
scientific article; zbMATH DE number 1438389