Topology, Calculus and Approximation

• Includes the theory of nets to study arbitrary topological spaces

• Illustrates the insufficiency of sequences with simple but rarely presented examples

• Elaborates on the historical background of most notions and results treated in this volume

Textbook

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

1. Front Matter
Pages i-xiv
2. Topology

1. Front Matter
Pages 1-2
2. Vilmos Komornik
Pages 3-35
3. Vilmos Komornik
Pages 37-64
4. Vilmos Komornik
Pages 65-93
3. Differential Calculus

1. Front Matter
Pages 95-96
2. Vilmos Komornik
Pages 97-116
3. Vilmos Komornik
Pages 117-140
4. Vilmos Komornik
Pages 141-163
5. Vilmos Komornik
Pages 165-190
4. Approximation Methods

1. Front Matter
Pages 191-191
2. Vilmos Komornik
Pages 193-217
3. Vilmos Komornik
Pages 219-229
4. Vilmos Komornik
Pages 231-266
5. Vilmos Komornik
Pages 267-282
6. Vilmos Komornik
Pages 283-297
5. Back Matter
Pages 299-379

Introduction

Presenting basic results of topology, calculus of several variables, and approximation theory which are rarely treated in a single volume, this textbook includes several beautiful, but almost forgotten, classical theorems of Descartes, Erdős, Fejér, Stieltjes, and Turán.

The exposition style of Topology, Calculus and Approximation follows the Hungarian mathematical tradition of Paul Erdős and others. In the first part, the classical results of Alexandroff, Cantor, Hausdorff, Helly, Peano, Radon, Tietze and Urysohn illustrate the theories of metric, topological and normed spaces. Following this, the general framework of normed spaces and Carathéodory's definition of the derivative are shown to simplify the statement and proof of various theorems in calculus and ordinary differential equations. The third and final part is devoted to interpolation, orthogonal polynomials, numerical integration, asymptotic expansions and the numerical solution of algebraic and differential equations.

Students of both pure and applied mathematics, as well as physics and engineering should find this textbook useful. Only basic results of one-variable calculus and linear algebra are used, and simple yet pertinent examples and exercises illustrate the usefulness of most theorems. Many of these examples are new or difficult to locate in the literature, and so the original sources of most notions and results are given to help readers understand the development of the field.

Keywords

metric space normed space topology calculus differential equations implicit functions interpolation orthogonal polynomials numerical integration Sturm sequences Newton's method Runge-Kutta method

Authors and affiliations

1. 1.Department of MathematicsUniversity of StrasbourgStrasbourgFrance

Vilmos Komornik has studied in Budapest, Hungary, and has taught in Hungary and France for nearly 40 years. His main research fields are control theory of partial differential equations and combinatorial number theory. He has made a number of contributions to the theory of J.L. Lions on exact controllability and stabilization and has co-authored several papers on expansions in noninteger bases with P. Erdős. He is an external member of the Hungarian Academy of Sciences.

Bibliographic information

• Book Title Topology, Calculus and Approximation
• Authors Vilmos Komornik
• Series Title Springer Undergraduate Mathematics Series
• Series Abbreviated Title SUMS
• DOI https://doi.org/10.1007/978-1-4471-7316-8
• Copyright Information Springer-Verlag London Ltd. 2017
• Publisher Name Springer, London
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-1-4471-7315-1
• eBook ISBN 978-1-4471-7316-8
• Series ISSN 1615-2085
• Series E-ISSN 2197-4144
• Edition Number 1
• Number of Pages XIV, 382
• Number of Illustrations 63 b/w illustrations, 1 illustrations in colour