Analysis in Vector Spaces (eBook)

MUSTAFA A. AKCOGLU, PhD, is Professor Emeritus in the Department of Mathematics at the University of Toronto, Canada. He has authored or coauthored over sixty journal articles on the topics of ergodic theory, functional analysis, and harmonic analysis. PAUL F.A. BARTHA, PhD, is Associate Professor in the Department of Philosophy at The University of British Columbia, Canada. He has authored or coauthored journal articles on topics such as probability and symmetry, probabilistic paradoxes, and the general philosophy of science. DZUNG MINH HA, PhD, is Associate Professor in the Department of Mathematics at Ryerson University, Canada. Dr. Ha focuses his research in the areas of ergodic theory and operator theory.

Preface.

PART I BACKGROUND MATERIAL.

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.

PART II DIFFERENTIATION.

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.

PART III INTEGRATION.

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.

PART IV APPENDICES.

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.

Index.

"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)