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Fundamentals of Advanced Mathematics V2 - 1st Edition - ISBN: 9781785482496, 9780081023853

Fundamentals of Advanced Mathematics V2

1st Edition

Field extensions, topology and topological vector spaces, functional spaces, and sheaves

Author: Henri Bourles
eBook ISBN: 9780081023853
Hardcover ISBN: 9781785482496
Imprint: ISTE Press - Elsevier
Published Date: 17th January 2018
Page Count: 360
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The three volumes of this series of books, of which this is the second, put forward the mathematical elements that make up the foundations of a number of contemporary scientific methods: modern theory on systems, physics and engineering.

Whereas the first volume focused on the formal conditions for systems of linear equations (in particular of linear differential equations) to have solutions, this book presents the approaches to finding solutions to polynomial equations and to systems of linear differential equations with varying coefficients.

Fundamentals of Advanced Mathematics, Volume 2: Field Extensions, Topology and Topological Vector Spaces, Functional Spaces, and Sheaves begins with the classical Galois theory and the theory of transcendental field extensions. Next, the differential side of these theories is treated, including the differential Galois theory (Picard-Vessiot theory of systems of linear differential equations with time-varying coefficients) and differentially transcendental field extensions. The treatment of analysis includes topology (using both filters and nets), topological vector spaces (using the notion of disked space, which simplifies the theory of duality), and the radon measure (assuming that the usual theory of measure and integration is known).

In addition, the theory of sheaves is developed with application to the theory of distributions and the theory of hyperfunctions (assuming that the usual theory of functions of the complex variable is known). This volume is the prerequisite to the study of linear systems with time-varying coefficients from the point-of-view of algebraic analysis and the algebraic theory of nonlinear systems.

Key Features

  • Present Galois Theory, transcendental field extensions, and Picard
  • Includes sections on Vessiot theory, differentially transcendental field extensions, topology, topological vector spaces, Radon measure, differential calculus in Banach spaces, sheaves, distributions, hyperfunctions, algebraic analysis, and local analysis of systems of linear differential equations


Graduate students in Systems Theory, Robotics, Physics or Mathematics, research engineers in Automation control and/or robotics, assistant professors and professors in Automation control and/or robotics

Table of Contents

1. Field Extensions and Differential Field Extensions
2. General Topology
3. Topological Vector Spaces
4. Measure and Integration, Function Spaces
5. Sheaves


No. of pages:
© ISTE Press - Elsevier 2018
17th January 2018
ISTE Press - Elsevier
eBook ISBN:
Hardcover ISBN:

About the Author

Henri Bourles

Henri Bourlès is Full Professor and Chair at the Conservatoire National des Arts et Métiers, Paris, France.

Affiliations and Expertise

Conservatoire National des Arts et Metiers, France


"As with the first volume, I’m not sure I see this book having extensive use as a textbook (not only because of the succinct exposition, but because the array of topics covered doesn’t match up with any standard course at the graduate level). However, also as with the first edition, the considerable amount of material included here and the efficient, concise way in which it is presented makes this book valuable as a reference." --MAA Reviews

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