Fundamental Concepts in Modern Analysis

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.
Contents: Preface to the Second EditionPreface to the First EditionPreliminary NotionsIt Began with the NumbersBasic Concepts in TopologyAdvanced Concepts in TopologyMetric Structures in Vector SpacesDifferentiation in Normed Vector SpacesHigher Order DerivativesDifferentiable ManifoldsAn Introduction to Singularity TheoryAn Introduction to Geometric Variational ProblemsBibliographyList of SymbolsIndex
Readership: Lecturers and students in pure and applied mathematics, theoretical engineering and the physical sciences.General Topology;Differentiability in Normed Vector Spaces;Differentiable Manifolds;Singularity Theory;Optimization Theory;Global Analysis0Key Features:The book starts from the basic notions of continuity and gradually builds up the theory of topological spaces, so that the reader goes from the intuitive notions to the abstract notions with as little pain as possibleSimilarly, differentiability is gradually built up from the intuitive to the abstract notions. Full proofs are given for the Chain Rule and the Inverse Function Theorem in Banach spacesFew books fill the gap from mathematical modeling based on elementary topology and calculus in finite dimensional real number spaces, to contemporary mathematics where advanced methods from topology and analysis pave the way for application of new strong tools such as manifolds and infinite dimensional function spaces. The present book attempts to fill this gap