Fundamental Concepts In Modern Analysis: An Introduction To Nonlinear Analysis (Second Edition)

by Vagn Lundsgaard Hansen, Poul G Hjorth

Overview

Many applied mathematical disciplines, such as dynamical systems and optimization theory as well as classical mathematical disciplines like differential geometry and the theory of Lie groups, have a common foundation in general topology and multivariate calculus in normed vector spaces. In this book, students from both pure and applied subjects are offered an opportunity to work seriously with fundamental notions from mathematical analysis that are important not only from a mathematical point of view but also occur frequently in the theoretical parts of, for example, the engineering sciences. The book provides complete proofs of the basic results from topology and differentiability of mappings in normed vector spaces. It is a useful resource for students and researchers in mathematics and the many sciences that depend on fundamental techniques from mathematical analysis.In this second edition, the notions of compactness and sequentially compactness are developed with independent proofs for the main results. Thereby the material on compactness is apt for direct applications also in functional analysis, where the notion of sequentially compactness prevails. This edition also covers a new section on partial derivatives, and new material has been incorporated to make a more complete account of higher order derivatives in Banach spaces, including full proofs for symmetry of higher order derivatives and Taylor's formula. The exercise material has been reorganized from a collection of problem sets at the end of the book to a section at the end of each chapter with further results. Readers will find numerous new exercises at different levels of difficulty for practice.

Product Details

ISBN-13:	9789811209406
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/28/2019
Pages:	304
Product dimensions:	6.00(w) x 9.00(h) x 0.69(d)

Table of Contents

Preface to the Second Edition; Preface to the First Edition; Preliminary Notions; It Began with the Numbers; Basic Concepts in Topology; Advanced Concepts in Topology; Metric Structures in Vector Spaces; Differentiation in Normed Vector Spaces; Higher Order Derivatives; Differentiable Manifolds; An Introduction to Singularity Theory; An Introduction to Geometric Variational Problems; Bibliography; List of Symbols; Index