Korevaar J. Mathematical Methods

New York, London: Academic Press, 1968. — X, 505 p.Volume 2 of this course, unfortunately, never appeared!Mathematical Methods, Volume 1: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions. The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector spaces, and delta sequences and convergence and approximation. The text then examines the Lebesgue integral, including approximation of integrable functions and applications, integration of sequences and series, functions of bounded variation and the Stieltjes integral, and multiple integrals. Curves and integrals, holomorphic functions and integrals in the complex plane, and multiple integrals are also discussed. The book is a valuable reference for students in the physical sciences, mathematics students interested in applications, and mathematically oriented engineering students.Rigorous but not abstract, this intensive introductory treatment provides many of the advanced mathematical tools used in applications. It also supplies the theoretical background that makes most other parts of modern mathematical analysis accessible. Geared toward advanced undergraduates and graduate students in the physical sciences and applied mathematics.The volumes on mathematical methods are intended for students in the physical sciences, for mathematics students with an interest in applications, and for mathematically oriented engineering students. It has been the author's aim to provide
(1) Many of the advanced mathematical tools used in applications;
(2) A certain theoretical mathematical background that will make most other parts of modern mathematical analysis accessible to the student of physical science, and that will make it easier for him to keep up with future mathematical developments in his field.From a mathematical point of view the presentation is fairly rigorous, but certainly not abstract. If the student of physical science finds parts of the material somewhat "theoretical," he should realize that the power of modern mathematics derives in large measure from its abstractness: It is because of its generality that mathematics is so widely applicable. If the student of mathematics finds parts of the material somewhat concrete, he should realize that mathematics is useful largely because it enables one to make calculations.Algebraic Theory of Vector Spaces
Introduction to Functional Analysis. Distributions
The Lebesgue Integral and Related Topics