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.

• Analysis in Vector SpacesPrefacePART I BACKGROUND MATERIAL1 Sets and Functions1.1 Sets in General1.2 Sets of Numbers1.3 Functions2 Real Numbers2.1 Review of the Order Relations2.2 Completeness of Real Numbers2.3 Sequences of Real Numbers2.4 Subsequences2.5 Series of Real Numbers2.6 Intervals and Connected Sets3 Vector Functions3.1 Vector Spaces: The Basics3.2 Bilinear Functions3.3 Multilinear Functions3.4 Inner Products3.5 Orthogonal Projections3.6 Spectral TheoremPART II DIFFERENTIATION4 Normed Vector Spaces4.1 Preliminaries4.2 Convergence in Normed Spaces4.3 Norms of Linear and Multilinear Transformations4.4 Continuity in Normed Spaces4.5 Topology of Normed Spaces5 Derivatives5.1 Functions of a Real Variable5.2 Differentiable Functions5.3 Existence of Derivatives5.4 Partial Derivatives5.5 Rules of Differentiation5.6 Differentiation of Products6 Diffeomorphisms and Manifolds6.1 The Inverse Function Theorem6.2 Graphs6.3 Manifolds in Parametric Representations6.4 Manifolds in Implicit Representations6.5 Differentiation on Manifolds7 Higher-Order Derivatives7.1 Definitions7.2 Change of Order in Differentiation7.3 Sequences of Polynomials7.4 Local Extremal ValuesPART III INTEGRATION8 Multiple Integrals8.1 Jordan Sets and Volume8.2 Integrals8.3 Images of Jordan Sets8.4 Change of Variables9 Integration on Manifolds9.1 Euclidean Volumes9.2 Integration on Manifolds9.3 Oriented Manifolds9.4 Integrals of Vector Fields9.5 Integrals of Tensor Fields9.6 Integration on Graphs10 Stokes' Theorem10.1 Basic Stokes' Theorem10.2 Flows10.3 Flux and Change of Volume in a Flow10.4 Exterior Derivatives10.5 Regular and Almost Regular Sets10.6 Stokes' Theorem on ManifoldsPART IV APPENDICESAppendix A: Construction of the Real NumbersA.1 Field and Order Axioms in QA.2 Equivalence Classes of Cauchy Sequences in QA.3 Completeness of RAppendix B: Dimension of a Vector SpaceB.1 Bases and Linearly Independent SubsetsAppendix C: DeterminantsC.1 PermutationsC.2 Determinants of Square MatricesC.3 Determinant FunctionsC.4 Determinant of a Linear TransformationC.5 Determinants on Cartesian ProductsC.6 Determinants in Euclidean SpacesC.7 Trace of an OperatorAppendix D: Partitions of UnityD.1 Partitions of UnityIndexSolutions Manual to accompany Analysis in Vector SpacesPreface ix1 Sets and Functions 32 Real Numbers 133 Vector Functions 274 Normed Vector Spaces 455 Derivatives 636 Diffeomorphisms and Manifolds 837 High-Order Derivatives 958 Multiple Integrals 999 Integration of Manifolds 11510 Stokes Theorem 131

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.

• Analysis in Vector SpacesPrefacePART I BACKGROUND MATERIAL1 Sets and Functions1.1 Sets in General1.2 Sets of Numbers1.3 Functions2 Real Numbers2.1 Review of the Order Relations2.2 Completeness of Real Numbers2.3 Sequences of Real Numbers2.4 Subsequences2.5 Series of Real Numbers2.6 Intervals and Connected Sets3 Vector Functions3.1 Vector Spaces: The Basics3.2 Bilinear Functions3.3 Multilinear Functions3.4 Inner Products3.5 Orthogonal Projections3.6 Spectral TheoremPART II DIFFERENTIATION4 Normed Vector Spaces4.1 Preliminaries4.2 Convergence in Normed Spaces4.3 Norms of Linear and Multilinear Transformations4.4 Continuity in Normed Spaces4.5 Topology of Normed Spaces5 Derivatives5.1 Functions of a Real Variable5.2 Differentiable Functions5.3 Existence of Derivatives5.4 Partial Derivatives5.5 Rules of Differentiation5.6 Differentiation of Products6 Diffeomorphisms and Manifolds6.1 The Inverse Function Theorem6.2 Graphs6.3 Manifolds in Parametric Representations6.4 Manifolds in Implicit Representations6.5 Differentiation on Manifolds7 Higher-Order Derivatives7.1 Definitions7.2 Change of Order in Differentiation7.3 Sequences of Polynomials7.4 Local Extremal ValuesPART III INTEGRATION8 Multiple Integrals8.1 Jordan Sets and Volume8.2 Integrals8.3 Images of Jordan Sets8.4 Change of Variables9 Integration on Manifolds9.1 Euclidean Volumes9.2 Integration on Manifolds9.3 Oriented Manifolds9.4 Integrals of Vector Fields9.5 Integrals of Tensor Fields9.6 Integration on Graphs10 Stokes' Theorem10.1 Basic Stokes' Theorem10.2 Flows10.3 Flux and Change of Volume in a Flow10.4 Exterior Derivatives10.5 Regular and Almost Regular Sets10.6 Stokes' Theorem on ManifoldsPART IV APPENDICESAppendix A: Construction of the Real NumbersA.1 Field and Order Axioms in QA.2 Equivalence Classes of Cauchy Sequences in QA.3 Completeness of RAppendix B: Dimension of a Vector SpaceB.1 Bases and Linearly Independent SubsetsAppendix C: DeterminantsC.1 PermutationsC.2 Determinants of Square MatricesC.3 Determinant FunctionsC.4 Determinant of a Linear TransformationC.5 Determinants on Cartesian ProductsC.6 Determinants in Euclidean SpacesC.7 Trace of an OperatorAppendix D: Partitions of UnityD.1 Partitions of UnityIndexSolutions Manual to accompany Analysis in Vector SpacesPreface ix1 Sets and Functions 32 Real Numbers 133 Vector Functions 274 Normed Vector Spaces 455 Derivatives 636 Diffeomorphisms and Manifolds 837 High-Order Derivatives 958 Multiple Integrals 999 Integration of Manifolds 11510 Stokes Theorem 131

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.

• Analysis in Vector SpacesPrefacePART I BACKGROUND MATERIAL1 Sets and Functions1.1 Sets in General1.2 Sets of Numbers1.3 Functions2 Real Numbers2.1 Review of the Order Relations2.2 Completeness of Real Numbers2.3 Sequences of Real Numbers2.4 Subsequences2.5 Series of Real Numbers2.6 Intervals and Connected Sets3 Vector Functions3.1 Vector Spaces: The Basics3.2 Bilinear Functions3.3 Multilinear Functions3.4 Inner Products3.5 Orthogonal Projections3.6 Spectral TheoremPART II DIFFERENTIATION4 Normed Vector Spaces4.1 Preliminaries4.2 Convergence in Normed Spaces4.3 Norms of Linear and Multilinear Transformations4.4 Continuity in Normed Spaces4.5 Topology of Normed Spaces5 Derivatives5.1 Functions of a Real Variable5.2 Differentiable Functions5.3 Existence of Derivatives5.4 Partial Derivatives5.5 Rules of Differentiation5.6 Differentiation of Products6 Diffeomorphisms and Manifolds6.1 The Inverse Function Theorem6.2 Graphs6.3 Manifolds in Parametric Representations6.4 Manifolds in Implicit Representations6.5 Differentiation on Manifolds7 Higher-Order Derivatives7.1 Definitions7.2 Change of Order in Differentiation7.3 Sequences of Polynomials7.4 Local Extremal ValuesPART III INTEGRATION8 Multiple Integrals8.1 Jordan Sets and Volume8.2 Integrals8.3 Images of Jordan Sets8.4 Change of Variables9 Integration on Manifolds9.1 Euclidean Volumes9.2 Integration on Manifolds9.3 Oriented Manifolds9.4 Integrals of Vector Fields9.5 Integrals of Tensor Fields9.6 Integration on Graphs10 Stokes' Theorem10.1 Basic Stokes' Theorem10.2 Flows10.3 Flux and Change of Volume in a Flow10.4 Exterior Derivatives10.5 Regular and Almost Regular Sets10.6 Stokes' Theorem on ManifoldsPART IV APPENDICESAppendix A: Construction of the Real NumbersA.1 Field and Order Axioms in QA.2 Equivalence Classes of Cauchy Sequences in QA.3 Completeness of RAppendix B: Dimension of a Vector SpaceB.1 Bases and Linearly Independent SubsetsAppendix C: DeterminantsC.1 PermutationsC.2 Determinants of Square MatricesC.3 Determinant FunctionsC.4 Determinant of a Linear TransformationC.5 Determinants on Cartesian ProductsC.6 Determinants in Euclidean SpacesC.7 Trace of an OperatorAppendix D: Partitions of UnityD.1 Partitions of UnityIndexSolutions Manual to accompany Analysis in Vector SpacesPreface ix1 Sets and Functions 32 Real Numbers 133 Vector Functions 274 Normed Vector Spaces 455 Derivatives 636 Diffeomorphisms and Manifolds 837 High-Order Derivatives 958 Multiple Integrals 999 Integration of Manifolds 11510 Stokes Theorem 131