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Analysis in Vector Spaces



Analysis in Vector Spaces

Mustafa A. Akcoglu, Paul F. A. Bartha, Dzung Minh Ha

ISBN: 978-1-118-16459-4 September 2011 480 Pages

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A rigorous introduction to calculus in vector spaces

The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences.

The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes:

  • Sets and functions

  • Real numbers

  • Vector functions

  • Normed vector spaces

  • First- and higher-order derivatives

  • Diffeomorphisms and manifolds

  • Multiple integrals

  • Integration on manifolds

  • Stokes' theorem

  • Basic point set topology

Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants.

Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.



1 Sets and Functions.

1.1 Sets in General.

1.2 Sets of Numbers.

1.3 Functions.

2 Real Numbers.

2.1 Review of the Order Relations.

2.2 Completeness of Real Numbers.

2.3 Sequences of Real Numbers.

2.4 Subsequences.

2.5 Series of Real Numbers.

2.6 Intervals and Connected Sets.

3 Vector Functions.

3.1 Vector Spaces: The Basics.

3.2 Bilinear Functions.

3.3 Multilinear Functions.

3.4 Inner Products.

3.5 Orthogonal Projections.

3.6 Spectral Theorem.


4 Normed Vector Spaces.

4.1 Preliminaries.

4.2 Convergence in Normed Spaces.

4.3 Norms of Linear and Multilinear Transformations.

4.4 Continuity in Normed Spaces.

4.5 Topology of Normed Spaces.

5 Derivatives.

5.1 Functions of a Real Variable.

5.2 Differentiable Functions.

5.3 Existence of Derivatives.

5.4 Partial Derivatives.

5.5 Rules of Differentiation.

5.6 Differentiation of Products.

6 Diffeomorphisms and Manifolds.

6.1 The Inverse Function Theorem.

6.2 Graphs.

6.3 Manifolds in Parametric Representations.

6.4 Manifolds in Implicit Representations.

6.5 Differentiation on Manifolds.

7 Higher-Order Derivatives.

7.1 Definitions.

7.2 Change of Order in Differentiation.

7.3 Sequences of Polynomials.

7.4 Local Extremal Values.


8 Multiple Integrals.

8.1 Jordan Sets and Volume.

8.2 Integrals.

8.3 Images of Jordan Sets.

8.4 Change of Variables.

9 Integration on Manifolds.

9.1 Euclidean Volumes.

9.2 Integration on Manifolds.

9.3 Oriented Manifolds.

9.4 Integrals of Vector Fields.

9.5 Integrals of Tensor Fields.

9.6 Integration on Graphs.

10 Stokes’ Theorem.

10.1 Basic Stokes’ Theorem.

10.2 Flows.

10.3 Flux and Change of Volume in a Flow.

10.4 Exterior Derivatives.

10.5 Regular and Almost Regular Sets.

10.6 Stokes’ Theorem on Manifolds.


Appendix A: Construction of the Real Numbers.

A.1 Field and Order Axioms in Q.

A.2 Equivalence Classes of Cauchy Sequences in Q.

A.3 Completeness of R.

Appendix B: Dimension of a Vector Space.

B.1 Bases and Linearly Independent Subsets.

Appendix C: Determinants.

C.1 Permutations.

C.2 Determinants of Square Matrices.

C.3 Determinant Functions.

C.4 Determinant of a Linear Transformation.

C.5 Determinants on Cartesian Products.

C.6 Determinants in Euclidean Spaces.

C.7 Trace of an Operator.

Appendix D: Partitions of Unity.

D.1 Partitions of Unity.


"The authors do not shy away from doing the hard work involved in proving say, the change of variable theorem for integration, the inverse function theorem, and Stokes's theorem--work which is not generally seen in standard calculus books--and thus they are quite correct when they state that students need a firm grip on single-variable calculus and some linear algebra, and a good comfort level with the comprehension and construction of rigorous proofs. Includes many examples and an excellent selection of exercises." (CHOICE, November 2010)
  • All proofs have been written with the student reader in mind.  In each proof, the notation and methods have been chosen to maximize clarity and transparency.
  • A large number of examples are included to reinforce new concepts and to illustrate how a result or technique can be applied to additional problems.  Counter-examples are also given throughout the book to warn the students against possible mistakes in the subject.
  • The book is self-contained, and the background material in the first few chapters is sufficient to enable well-motivated students to read the book without undue difficulties. 
  • To avoid possible ambiguities and misunderstandings, terminology and notation are carefully defined before being used.
  • An extensive collection of exercises is provided, and full solutions are included in an accompanying Student Solutions Manual.