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Analysis in Vector Spaces

Mustafa A. Akcoglu, Paul F. A. Bartha, and Dzung Minh Ha
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
David S. Mazel
, on

The first time I encountered analysis was in graduate school taking a course in real analysis. Such a course would teach students, as one friend put it, “how to write a proof.” That was quite true. From this book I found that what I was really taught might better be called scalar analysis because we worked only with scalar functions. It didn’t occur to me that the same analysis techniques of scalars would apply to vector functions. There is much more to analysis than scalars, as this book can attest.

For those of us who seek to expand our experiences, and to learn analysis on vector spaces, this book is a good start. The format is “classic analysis textbook”: definitions; lemmas and proofs; and theorems (sometimes built from lemmas) and proofs. If that appeals to you, you won’t be disappointed.

The authors begin with a review of sets, numbers, and functions. The discussion is basic but it familiarizes the reader to the authors’ notation and is well worth the time. We then explore real numbers, convergent sequences, and linear transformations. It is with linear transformations that we begin to get to vector spaces and functions in vector spaces. I won’t go through the many sections because MAA provides the table of contents, please see the link above.

The book covers all its topics thoroughly and with examples that beautifully illustrate the ideas. For example, the authors define and explain rigid motion, trajectories, and the Frenet formulas (unit tangent vector, unit principal normal vector, and binormal vector) so that we can explore curvature and torsion. As with much of the book, the discussion is theoretical within the text, but the problems provide the reader with concrete insights. The problems take the material to a higher level.

If you haven’t studied these topics in vector spaces before, this book will serve you well.

David S. Mazel received his Ph. D. from Georgia Tech in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.




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.