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关键词：mathematics Mathematic Thematic Springer Graduate Series 电子版 经典

SUMS    Metric Spaces    Searcoid
There is no philosophy which is not founded upon knowledge of
the phenomena, but to get any profit from this knowledge
it is absolutely necessary
to be a mathematician. Daniel Bernoulli, 1700–1782
To the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Cumulative Reference Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1. Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Point Functions and Pointlike Functions . . . . . . . . . . . . . . . . . . . . 8
1.3 Metric Subspaces and Metric Superspaces . . . . . . . . . . . . . . . . . . 10
1.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Extending a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Metrics on Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Metrics and Norms on Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . 16
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Distances from Points to Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Inequalities for Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Distances to Unions and Intersections . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Isolated Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Accumulation Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Distances from Sets to Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Nearest Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4. Open, Closed and Dense Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 Open and Closed Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Dense Subsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Topologies on Subspaces and Superspaces . . . . . . . . . . . . . . . . . . 61
4.5 Topologies on Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6 Universal Openness and Universal Closure . . . . . . . . . . . . . . . . . . 64
4.7 Nests of Closed Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5. Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 Open and Closed Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Using Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Balls in Subspaces and in Products . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Balls in Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6. Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1 Definition of Convergence for Sequences . . . . . . . . . . . . . . . . . . . . 83
6.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Superior and Inferior Limits of Real Sequences . . . . . . . . . . . . . . 86
6.4 Convergence in Subspaces and Superspaces . . . . . . . . . . . . . . . . . 88
6.5 Convergence in Product Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.6 Convergence Criteria for Interior and Closure . . . . . . . . . . . . . . . 90
6.7 Convergence of Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.8 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.9 Cauchy Sequences in Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
13. Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
13.1 Topological Equivalence of Metrics. . . . . . . . . . . . . . . . . . . . . . . . . 227
13.2 Uniform Equivalence of Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
13.3 Lipschitz Equivalence of Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
13.4 The Truth about Conserving Metrics. . . . . . . . . . . . . . . . . . . . . . . 238
13.5 Equivalence of Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
13.6 Equivalent Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Appendix A. Language and Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
A.1 Theorems and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
A.2 Truth of Compound Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
A.3 If, and Only If . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
A.4 Transitivity of Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
A.5 Proof by Counterexample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
A.6 Vacuous Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
A.7 Proof by Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
A.8 Proof by Contraposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
A.9 Proof by Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
A.10 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
A.11 Let and Suppose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Appendix B. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
B.1 Notation for Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
B.2 Subsets and Supersets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
B.3 Universal Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
B.4

SUMS Linear Functional Analysis 2ed Bryan P. Rynne (2008 )
This book provides an introduction to the ideas and methods of linear functional
analysis at a level appropriate to the final year of an undergraduate
course at a British university. The prerequisites for reading it are a standard
undergraduate knowledge of linear algebra and real analysis (including the theory
of metric spaces).
Part of the development of functional analysis can be traced to attempts
to find a suitable framework in which to discuss differential and integral
equations. Often, the appropriate setting turned out to be a vector space of
real or complex-valued functions defined on some set. In general, such a vector
space is infinite-dimensional. This leads to difficulties in that, although
many of the elementary properties of finite-dimensional vector spaces hold in
infinite-dimensional vector spaces, many others do not. For example, in general
infinite-dimensional vector spaces there is no framework in which to make sense
of analytic concepts such as convergence and continuity. Nevertheless, on the
spaces of most interest to us there is often a norm (which extends the idea of
the length of a vector to a somewhat more abstract setting). Since a norm on a
vector space gives rise to a metric on the space, it is now possible to do analysis
in the space. As real or complex-valued functions are often called functionals,
the term functional analysis came to be used for this topic.
We now briefly outline the contents of the book. In Chapter 1 we present
(for reference and to establish our notation) various basic ideas that will be required
throughout the book. Specifically, we discuss the results from elementary
linear algebra and the basic theory of metric spaces which will be required in
later chapters. We also give a brief summary of the elements of the theory of
Lebesgue measure and integration. Of the three topics discussed in this introductory
chapter, Lebesgue integration is undoubtedly the most technically difficult
and the one which the prospective reader is least likely to have encountered

1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Lebesgue Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2. Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1 Examples of Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Finite-dimensional Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3. Inner Product Spaces, Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Orthonormal Bases in Infinite Dimensions . . . . . . . . . . . . . . . . . . . 72
3.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4. Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1 Continuous Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 The Norm of a Bounded Linear Operator . . . . . . . . . . . . . . . . . . . . 96
4.3 The Space B(X, Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Inverses of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5. Duality and the Hahn–Banach Theorem . . . . . . . . . . . . . . . . . . . . 121
5.1 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Sublinear Functionals, Seminorms and the Hahn–Banach
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Contents
5.3 The Hahn–Banach Theorem in Normed Spaces . . . . . . . . . . . . . . . 132
5.4 The General Hahn–Banach theorem . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5 The Second Dual, Reflexive Spaces
and Dual Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6 Projections and Complementary Subspaces . . . . . . . . . . . . . . . . . . 155
5.7 Weak and Weak-∗ Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6. Linear Operators on Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1 The Adjoint of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2 Normal, Self-adjoint and Unitary Operators . . . . . . . . . . . . . . . . . . 176
6.3 The Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.4 Positive Operators and Projections . . . . . . . . . . . . . . . . . . . . . . . . . 192
7. Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.1 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.2 Spectral Theory of Compact Operators . . . . . . . . . . . . . . . . . . . . . . 216
7.3 Self-adjoint Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8. Integral and Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.1 Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.2 Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.3 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.4 Eigenvalue Problems and Green’s Functions. . . . . . . . . . . . . . . . . . 253
9. Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

SUMS General Relativity (N.M.J. Woodhouse)
It is a challenging but rewarding task to teach general relativity to undergraduates.
Time and experience are in short supply. One can rely neither on the
undivided attention of students who are studying many other exciting topics
in the final years of their course, nor on easy familiarity with the classical
tools of applied mathematics and geometry. Not only are the ideas themselves
difficult, but the calculations needed to solve even quite simple problems are
themselves technically challenging for students who have only recently learned
about multivariable calculus and partial differential equations.
For those with a strong background in pure mathematics, there is the temptation
to present the theory as an application of differential geometry without
conveying a clear understanding of its detailed connection with physical observation.
At the other extreme, one can focus too exclusively on physical prediction,
and ask the audience to take too much of the mathematical argument on
trust.
This book is based on a course given at the Mathematical Institute in Oxford
over many years to final-year mathematics students. It is in the tradition
of physical applied mathematics as it is taught in this country, and may, I hope,
be of use elsewhere. It is coloured by the mathematical leaning of our students,
but does not present general relativity as a branch of differential geometry. The
geometric ideas, which are of course central to the understanding of the nature
of gravity, are introduced in parallel with the development of the theory—the
emphasis being on laying bare how one is led to pseudo-Riemannian geometry
through a natural process of reconciliation of special relativity with the equivalence
principle. At centre stage are the ‘local inertial coordinates’ set up by
an observer in free-fall, in which special relativity is valid over short times and
distances.
In more practical terms, the book is a sequel, with some overlap in


Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1. Newtonian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 ‘Special’ and ‘General’ Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Newton’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Gravity and Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 The Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Linearity and Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 The Starting Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. Inertial Coordinates and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Inertial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Tensors in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Operations on Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3. Energy-Momentum Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Electromagnetic Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . 37
4. Curved Space–Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Local Inertial Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Existence of Local Inertial Coordinates . . . . . . . . . . . . . . . . . . . . . 46
4.3 Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Null Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52



4.5 Transformation of the Christoffel Symbols . . . . . . . . . . . . . . . . . . . 53
4.6 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.7 Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.8 The Geometry of Surfaces* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.9 Summary of the Mathematical Formulation . . . . . . . . . . . . . . . . . . 64
5. Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1 The Derivative of a Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Covariant Derivatives of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 The Wave Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.7 Symmetries of the Riemann Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.8 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.9 Geodesic Triangles*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6. Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Tidal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 The Weak Field Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3 The Nonvacuum Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7. Spherical Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.1 The Field of a Static Spherical Body . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 The Curvature Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Stationary Observers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.4 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.5 Photons and Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . 101


12.1 Retarded Time in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.2 Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
12.3 Homogeneous and Isotropic Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 163
12.4 Cosmological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.5 Homogeneity in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
12.6 Cosmological Redshift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
12.7 Cosmological Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Appendix A: Notes on Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Appendix B: Further problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

SUMS Hyperbolic geometry
Preamble to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Preamble to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1. The Basic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Model for the Hyperbolic Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Riemann Sphere C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The Boundary at Infinity of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2. The General M¨obius Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 The Group of M¨obius Transformations . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Transitivity Properties of M¨ob+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 The Cross Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Classification of M¨obius Transformations . . . . . . . . . . . . . . . . . . . . 39
2.5 A Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Reflections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7 The Conformality of Elements of M¨ob. . . . . . . . . . . . . . . . . . . . . . . 53
2.8 Preserving H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9 Transitivity Properties of M¨ob(H) . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.10 The Geometry of the Action of M¨ob(H) . . . . . . . . . . . . . . . . . . . . . 65

3. Length and Distance in H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1 Paths and Elements of Arc-length . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 The Element of Arc-length on H . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3 Path Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 From Arc-length to Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5 Formulae for Hyperbolic Distance in H . . . . . . . . . . . . . . . . . . . . . . 99
3.6 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.7 Metric Properties of (H, dH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4. Planar Models of the Hyperbolic Plane . . . . . . . . . . . . . . . . . . . . . 117
4.1 The Poincar´e Disc Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2 A General Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5. Convexity, Area, and Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2 Hyperbolic Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3 The Definition of Hyperbolic Area . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.4 Area and the Gauss–Bonnet Formula . . . . . . . . . . . . . . . . . . . . . . . 169
5.5 Applications of the Gauss–Bonnet Formula . . . . . . . . . . . . . . . . . . 174
5.6 Trigonometry in the Hyperbolic Plane. . . . . . . . . . . . . . . . . . . . . . . 181
6. Nonplanar models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.1 The Hyperboloid Model of the Hyperbolic Plane . . . . . . . . . . . . . 189
6.2 Higher Dimensional Hyperbolic Spaces . . . . . . . . . . . . . . . . . . . . . . 209
Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
List of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

SUMS Game Theory Decisions Interaction And Evolution (2007)
This book is an introduction to game theory from a mathematical perspective.
It is intended to be a first course for undergraduate students of mathematics,
but I also hope that it will contain something of interest to advanced students
or researchers in biology and economics who often encounter the basics of game
theory informally via relevant applications. In view of the intended audience,
the examples used in this book are generally abstract problems so that the
reader is not forced to learn a great deal of a subject – either biology or economics
– that may be unfamiliar. Where a context is given, these are usually
“classical” problems of the subject area and are, I hope, easy enough to follow.
The prerequisites are generally modest. Apart from a familiarity with (or
a willingness to learn) the concepts of a proof and some mathematical notation,
the main requirement is an elementary understanding of probability. A
familiarity with basic calculus would be useful for Chapter 6 and some parts of
Chapters 1 and 8. The basic ideas of simple ordinary differential equations are
required in Chapter 9 and, towards the end of that chapter, some familiarity
with matrices would be an advantage – although the relevant ideas are briefly
described in an appendix.
I have tried to provide a unified account of single-person decision problems
(“games against nature”) as well as both classical and evolutionary game theory,
whereas most textbooks cover only one of these. There are two immediate
consequences of this broad approach. First, many interesting topics are left out.
However, I hope that this book will provide a good foundation for further study
and that the books suggested for further reading at the end of this volume will
go some way to filling the gaps. Second, the notation and terminology used
may be different in places from that which is commonly used in each of the
three separate areas. In this book, I have tried to use similar (combinations of)


Part I. Decisions
1. Simple Decision Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Optimisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Making Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Modelling Rational Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Modelling Natural Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Optimal Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2. Simple Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Strategic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Randomising Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Optimal Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3. Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 State-dependent Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Stochastic Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Optimal Strategies for Finite Processes . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Infinite-horizon Markov Decision Processes . . . . . . . . . . . . . . . . . . 48
3.6 Optimal Strategies for Infinite Processes . . . . . . . . . . . . . . . . . . . . . 50
3.7 Policy Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Part II. Interaction
4. Static Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Interactive Decision Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Describing Static Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Solving Games Using Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Existence of Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 The Problem of Multiple Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Classification of Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.8 Games with n-players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5. Finite Dynamic Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1 Game Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Information Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4 Behavioural Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5 Subgame Perfection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6 Nash Equilibrium Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6. Games with Continuous Strategy Sets . . . . . . . . . . . . . . . . . . . . . . 107
6.1 Infinite Strategy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 The Cournot Duopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 The Stackelberg Duopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 War of Attrition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7. Infinite Dynamic Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1 Repeated Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 The Iterated Prisoners’ Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3 Subgame Perfection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Folk Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.5 Stochastic Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Part III. Evolution
8. Population Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.1 Evolutionary

9. Replicator Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.1 Evolutionary Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.2 Two-strategy Pairwise Contests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.3 Linearisation and Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . 171
9.4 Games with More Than Two Strategies . . . . . . . . . . . . . . . . . . . . . 174
9.5 Equilibria and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Part IV. Appendixes
A. Constrained Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
B. Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

SMM Applied Proof Theory  Proof Interpretations and Their Use in Mathematics
Preface
This book gives an introduction to so-called proof interpretations, more specifically
various forms of realizability and functional interpretations, and their use in mathematics.
Whereas earlier treatments of these techniques (e.g. [366, 266, 122, 369, 7])
emphasize foundational and logical issues the focus of this book is on applications
of the methods to extract new effective information such as computable uniform
bounds from given (typically ineffective) proofs. This line of research, which has its
roots in G. Kreisel’s pioneering work on ‘unwinding of proofs’ from the 50’s, has
in more recent years developed into a field of mathematical logic which has been
called (suggested by D. Scott) ‘proof mining’. The areas where proof mining based
on proof interpretations has been applied most systematically are numerical analysis
and functional analysis and so the book concentrates on those. There are also some
extractions of effective information from proofs (guided by logic) in number theory
(G. Kreisel, H. Luckhardt, see e.g. [249, 268, 267, 122]) and algebra (G. Kreisel, C.
Delzell, H. Lombardi, T. Coquand and others, see e.g. [252, 84, 77, 74, 76]). However,
here mainly methods from structural proof theory such as Herbrand’s theorem,
ε -substitution and cut-elimination are used and we will refer to the literature for
more information on these results.
In this book two kinds of systems play an important role: those with full induction
and variants with induction for purely existential formulas (whose central role has
been singled out in the context of so-called reverse mathematics, [338]). Further
(still weaker) fragments are briefly discussed in comments and referred to in the
literature.
Modified realizability (due to G. Kreisel) and functional interpretation (due to K.
G¨odel) are both first developed in the framework of constructive (‘intuitionistic’)
arithmetic in higher types to which consecutively various non-constructive principles
are added.
After this, systems based on ordinary (‘classical’) logic are studied. It is shown
that the combination of G¨odel’s functional (‘Dialectica’) interpretation with the
so-called negative translation, which embeds certain classical theories into approximately
intuitionistic counterparts, can be used to unwind fully non-constructive
proofs. Since the main emphasis throughout this book is on ineffective proofs based
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Common Notations and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Unwinding proofs (‘Proof Mining’) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Introductory remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Informal treatment of ineffective proofs . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Herbrand’s theorem and the no-counterexample interpretation . . . . 22
2.4 Exercises, historical comments and suggested further reading . . . . 38
3 Intuitionistic and classical arithmetic in all finite types . . . . . . . . . . . . . 41
3.1 Intuitionistic and classical predicate logic . . . . . . . . . . . . . . . . . . . . . 41
3.2 Intuitionistic (‘Heyting’) arithmetic HA and Peano arithmetic PA . 44
3.3 Extensional intuitionistic (‘Heyting’) and classical (‘Peano’)
arithmetic in all finite types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Fragments of (W)E-HAω and (W)E-PAω . . . . . . . . . . . . . . . . . . . . . 52
3.5 Fragments corresponding to the Grzegorczyk hierarchy . . . . . . . . . . 54
3.6 Models of E-PAω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Exercises, historical comments and suggested further reading . . . . 73
4 RepresentationofPolish metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 Representation of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Representation of complete separable metric (‘Polish’) spaces . . . . 81
4.3 Special representation of compact metric spaces . . . . . . . . . . . . . . . . 88
4.4 Fragments, exercises, historical comments and suggested further
reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


12 A non-standard principle of uniform boundedness . . . . . . . . . . . . . . . . . 223
12.1 The Σ 0
1 -boundedness principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.2 Applications of Σ 0
1 -boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
12.3 Remarks on the fragments E-GnAω . . . . . . . . . . . . . . . . . . . . . . . . . . 238
12.4 Exercises, historical comments and suggested further reading . . . . 241
13 Elimination of monotone Skolem functions . . . . . . . . . . . . . . . . . . . . . . . . 243
13
15.2 Applications to uniqueness proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
15.3 Applications to monotone convergence theorems . . . . . . . . . . . . . . . 291
15.4 Applications to proofs of contractivity . . . . . . . . . . . . . . . . . . . . . . . . 292
15.5 Remarks on fragments of T ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
15.6 Historical comments and suggested further reading . . . . . . . . . . . . . 295
16 Case study I: Uniqueness proofs in approximation theory . . . . . . . . . . . 297
16.1 Uniqueness proofs in best approximation theory. . . . . . . . . . . . . . . . 297
16.2 Best Chebycheff approximation I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
16.3 Best Chebycheff approximation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
16.4 Best L1-approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
16.5 Exercises, historical comments and suggested further reading . . . . 376
17 Applications to analysis: general metatheorems II . . . . . . . . . . . . . . . . . 377
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
17.2 Main results in the metric and hyperbolic case . . . . . . . . . . . . . . . . . 391
17.3 The case
theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
18.4 Asymptotically nonexpansivemappings . . . . . . . . . . . . . . . . . . . . . . 496
18.5 Applications of proof mining in ergodic theory . . . . . . . . . . . . . . . . . 499
18.6 Exercises, historical comments and suggested further reading . . . . 501
19 Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
List of formal systems and term classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
List of axioms and rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

SMM Modern Methods in the Calculus of Variations  L^p spaces Fonseca G. Leoni
Preface
In recent years there has been renewed interest in the calculus of variations,
motivated in part by ongoing research in materials science and other disciplines.
Often, the study of certain material instabilities such as phase transitions,
formation of defects, the onset of microstructures in ordered materials,
fracture and damage, leads to the search for equilibria through a minimization
problem of the type
min {I (v) : v ∈ V},
where the class V of admissible functions v is a subset of some Banach space V .
This is the essence of the calculus of variations: the identification of necessary
and sufficient conditions on the functional I that guarantee the existence
of minimizers. These rest on certain growth, coercivity, and convexity conditions,
which often fail to be satisfied in the context of interesting applications,
thus requiring the relaxation of the energy. New ideas were needed, and the introduction
of innovative techniques has resulted in remarkable developments
in the subject over the past twenty years, somewhat scattered in articles,
preprints, books, or available only through oral communication, thus making
it difficult to educate young researchers in this area.
This is the first of two books in the calculus of variations and measure
theory in which many results, some now classical and others at the forefront
of research in the subject, are gathered in a unified, consistent way. A main
concern has been to use contemporary arguments throughout the text to revisit
and streamline well-known aspects of the theory, while providing novel
contributions.
The core of this book is the analysis of necessary and sufficient conditions
for sequential lower semicontinuity of functionals on Lp spaces, followed by
relaxation techniques. What sets this book apart from existing introductory
texts in the calculus of variations is twofold: Instead of laying down the theory
in the one-dimensional setting for integrands f = f(x, u, u
), we work in N
dimensions and no derivatives are present. In addition, it is self-contained in

Contents
Part I Measure Theory and Lp Spaces
1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Measures and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Measures and Outer Measures . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Radon and Borel Measures and Outer Measures . . . . . . . 22
1.1.3 Measurable Functions and Lebesgue Integration . . . . . . . 37
1.1.4 Comparison Between Measures . . . . . . . . . . . . . . . . . . . . . . 55
1.1.5 Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
1.1.6 Projection of Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . 83
1.2 Covering Theorems and Differentiation of Measures in RN . . . . 90
1.2.1 Covering Theorems in RN . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1.2.2 Differentiation Between Radon Measures in RN . . . . . . . 103
1.3 Spaces of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.3.1 Signed Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.3.2 Signed Finitely Additive Measures . . . . . . . . . . . . . . . . . . . 119
1.3.3 Spaces of Measures as Dual Spaces . . . . . . . . . . . . . . . . . . 123
1.3.4 Weak Star Convergence of Measures . . . . . . . . . . . . . . . . . 129
2 Lp Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.1 Abstract Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.1.1 Definition and Main Properties . . . . . . . . . . . . . . . . . . . . . . 139
2.1.2 Strong Convergence in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.1.3 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
2.1.4 Weak Convergence in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
2.1.5 Biting Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
2.2 Euclidean Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
2.2.1 Approximation by Regular Functions . . . . . . . . . . . . . . . . 190
2.2.2 Weak Convergence in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
2.2.3 Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
2.3 Lp Spaces on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.4 Sequential Lower Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . 436
6.4.1 Strong Convergence in Lp, 1 ≤ p < ∞. . . . . . . . . . . . . . . . 436
6.4.2 Strong Convergence in L∞ . . . . . . . . . . . . . . . . . . . . . . . . . 442
6.4.3 Weak Convergence in Lp, 1 ≤ p < ∞. . . . . . . . . . . . . . . . . 445
6.4.4 Weak Star Convergence in L∞ . . . . . . . . . . . . . . . . . . . . . . 448
6.4.5 Weak Star Convergence in the Sense of Measures . . . . . . 449
6.5 Integral Representation in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
6.6 Relaxation in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
6.6.1 Weak Convergence and Weak Star Convergence in Lp,
1 ≤ p≤∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
6.6.2 Weak Star Convergence in the Sense of Measures in L1 . 478
7 Integrands f = f (x, u, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
7.1 Convex Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
7.2 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
7.3 Sequential Lower Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . 491
7.3.1 Strong–Strong Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 491
7.3.2 Strong–Weak Convergence 1 ≤ p, q < ∞ . . . . . . . . . . . . . 492
7.4 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
8 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
8.1 The Fundamental Theorem for Young Measures . . . . . . . . . . . . . 518
8.2 Characterization of Young Measures . . . . . . . . . . . . . . . . . . . . . . . 532
8.2.1 The Homogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
8.2.2 The Inhomogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
8.3 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
Part IV Appendix
A Functional Analysis and Set Theory . . . . . . . . . . . . . . . . . . . . . . . 549
A.1 Some Results from Functional Analysis . . . . . . . . . . . . . . . . . . . . . 549
A.1.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
A.1.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
A.1.3 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 554
A.1.4 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
A.1.5 Weak Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
A.1.6 Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
A.1.7 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
A.2 Wellorderings, Ordinals, and Cardinals . . . . . . . . . . . . . . . . . . . . . 567
B Notes and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Notation and List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

SMM Methods in Nonlinear Analysis(Kung-Ching Chang)
Preface
Nonlinear analysis is a new area that was born and has matured from abundant
research developed in studying nonlinear problems. In the past thirty
years, nonlinear analysis has undergone rapid growth; it has become part of
the mainstream research fields in contemporary mathematical analysis.
Many nonlinear analysis problems have their roots in geometry, astronomy,
fluid and elastic mechanics, physics, chemistry, biology, control theory, image
processing and economics. The theories and methods in nonlinear analysis
stem from many areas of mathematics: Ordinary differential equations, partial
differential equations, the calculus of variations, dynamical systems, differential
geometry, Lie groups, algebraic topology, linear and nonlinear functional
analysis, measure theory, harmonic analysis, convex analysis, game theory,
optimization theory, etc. Amidst solving these problems, many branches are
intertwined, thereby advancing each other.
The author has been offering a course on nonlinear analysis to graduate
students at Peking University and other universities every two or three
years over the past two decades. Facing an enormous amount of material,
vast numbers of references, diversities of disciplines, and tremendously different
backgrounds of students in the audience, the author is always concerned
with how much an individual can truly learn, internalize and benefit from a
mere semester course in this subject.
The author’s approach is to emphasize and to demonstrate the most fundamental
principles and methods through important and interesting examples
from various problems in different branches of mathematics. However, there
are technical difficulties: Not only do most interesting problems require background
knowledge in other branches of mathematics, but also, in order to solve
these problems, many details in argument and in computation should be included.
In this case, we have to get around the real problem, and deal with a
simpler one, such that the application of the method is understandable. The
author does not always pursue each theory in its broadest generality; instead,
he stresses the motivation, the success in applications and its limitations.
Contents
1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Differential Calculus in Banach Spaces . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Frechet Derivatives and Gateaux Derivatives . . . . . . . . . . 2
1.1.2 Nemytscki Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.3 High-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Implicit Function Theorem and Continuity Method . . . . . . . . . . 12
1.2.1 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.3 Continuity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Lyapunov–Schmidt Reduction and Bifurcation . . . . . . . . . . . . . . 30
1.3.1 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.2 Lyapunov–Schmidt Reduction . . . . . . . . . . . . . . . . . . . . . . . 33
1.3.3 A Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.4 Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.5 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4 Hard Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.4.1 The Small Divisor Problem . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.4.2 Nash–Moser Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2 Fixed-Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.1 Order Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2 Convex Function and Its Subdifferentials . . . . . . . . . . . . . . . . . . . 80
2.2.1 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.2 Subdifferentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.3 Convexity and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4 Nonexpansive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.5 Monotone Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.6 Maximal Monotone Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.6 Free Discontinuous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.6.1 Γ-convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.6.2 A Phase Transition Problem . . . . . . . . . . . . . . . . . . . . . . . . 280
4.6.3 Segmentation and Mumford–Shah Problem . . . . . . . . . . . 284
4.7 Concentration Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.7.1 Concentration Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.7.2 The Critical Sobolev Exponent and the Best Constants 295
4.8 Minimax Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
4.8.1 Ekeland Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 301
4.8.2 Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
4.8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
5 Topological and Variational Methods . . . . . . . . . . . . . . . . . . . . . . 315
5.1 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1.2 Deformation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
5.1.3 Critical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
5.1.4 Global Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
5.1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
5.2 Minimax Principles (Revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
5.2.1 A Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
5.2.2 Category and Ljusternik–Schnirelmann
Multiplicity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
5.2.3 Cap Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
5.2.4 Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
5.2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
5.3 Periodic Orbits for Hamiltonian System
and Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
5.3.1 Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
5.3.2 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
5.3.3 Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
5.4 Prescribing Gaussian Curvature Problem on S2 . . . . . . . . . . . . . 380
5.4.1 The Conformal Group and the Best Constant . . . . . . . . . 380
5.4.2 The Palais–Smale Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 387
5.4.3 Morse Theory for the Prescribing Gaussian Curvature
Equation on S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
5.5 Conley Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
5.5.1 Isolated Invariant Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
5.5.2 Index Pair and Conley Index. . . . . . . . . . . . . . . . . . . . . . . . 397
5.5.3 Morse Decomposition on Compact Invariant Sets
and Its Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

SMM Normal Forms and Unfoldings for Local Dynamical Systems
Preface
The subject of local dynamical systems is concerned with the following two
questions:
1. Given an n × n matrix A, describe the behavior, in a neighborhood
of the origin, of the solutions of all systems of differential equations
having a rest point at the origin with linear part Ax, that is, all
systems of the form
x˙ = Ax + · · · ,
where x ∈ Rn and the dots denote terms of quadratic and higher
order.
2. Describe the behavior (near the origin) of all systems close to a system
of the type just described.
To answer these questions, the following steps are employed:
1. A normal form is obtained for the general system with linear part
Ax. The normal form is intended to be the simplest form into which
any system of the intended type can be transformed by changing the

Contents
Preface v
1 Two Examples 1
1.1 The (Single) Nonlinear Center . . . . . . . . . . . . . . . 1
1.2 The Nonsemisimple Double-Zero Eigenvalue . . . . . . . 21
2 The Splitting Problem for Linear Operators 27
2.1 The Splitting Problem in the Semisimple Case . . . . . . 28
2.2 Splitting by Means of an Inner Product . . . . . . . . . . 32
2.3 Nilpotent Operators . . . . . . . . . . . . . . . . . . . . . 34
2.4 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 * An Introduction to sl(2) Representation Theory . . . . 47
2.6 * Algorithms for the sl(2) Splittings . . . . . . . . . . . . 57
2.7 * Obtaining sl(2) Triads . . . . . . . . . . . . . . . . . . 63
3 Linear Normal Forms 69
3.1 Perturbations of Matrices . . . . . . . . . . . . . . . . . . 69
3.2 An Introduction to the Five Formats . . . . . . . . . . . 74
3.3 Normal and Hypernormal Forms When A0 Is Semisimple 87
3.4 Inner Product and Simplified Normal Forms . . . . . . . 99
3.5 * The sl(2) Normal Form . . . . . . . . . . . . . . . . . . 118
3.6 Lie Theory and the Generated Formats . . . . . . . . . . 129
3.7 Metanormal Forms and k-Determined Hyperbolicity . . . 142
xviii Contents
4 Nonlinear Normal Forms 157
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.2 Settings for Nonlinear Normal Forms . . . . . . . . . . . 160
4.3 The Direct Formats (1a and 1b) . . . . . . . . . . . . . . 164
4.4 Lie Theory and the Generated Formats . . . . . . . . . . 174
4.5 The Semisimple Normal Form . . . . . . . . . . . . . . . 190
4.6 The Inner Product and Simplified Normal Forms . . . . 221
4.7 The Module Structure of Inner Product and
Simplified Normal Forms . . . . . . . . . . . . . . . . . . 242
4.8 * The sl(2) Normal Form . . . . . . . . . . . . . . . . . . 265
4.9 The Hamiltonian Case . . . . . . . . . . . . . . . . . . . 271
4.10 Hypernormal Forms for Vector Fields . . . . . . . . . . . 283
5 Geometrical Structures in Normal Forms 295
5.1 Preserved Structures in Truncated Normal Forms . . . . 297
5.2 Geometrical Structures in the Full System . . . . . . . . 316
5.3 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . 323
5.4 The Nilpotent Center Manifold Case . . . . . . . . . . . 335
6 Selected Topics in Local Bifurcation Theory 339
6.1 Bifurcations from a Single-Zero Eigenvalue:
A “Neoclassical” Approach . . . . . . . . . . . . . . . . . 341
6.2 Bifurcations from a Single-Zero Eigenvalue:
A “Modern” Approach . . . . . . . . . . . . . . . . . . . 356
6.3 Unfolding the Single-Zero Eigenvalue . . . . . . . . . . . 366
6.4 Unfolding in the Presence of
Generic Quadratic Terms . . . . . . . . . . . . . . . . . . 371
6.5 Bifurcations from a Single Center (Hopf and
Degenerate Hopf Bifurcations) . . . . . . . . . . . . . . . 382
6.6 Bifurcations from the Nonsemisimple Double-Zero
Eigenvalue (Takens–Bogdanov Bifurcations) . . . . . . . 389
6.7 Unfoldings of Mode Interactions . . . . . . . . . . . . . . 396
A Rings 405
A.1 Rings, Ideals, and Division . . . . . . . . . . . . . . . . . 405
A.2 Monomials and Monomial Ideals . . . . . . . . . . . . . . 412
A.3 Flat Functions and Formal Power Series . . . . . . . . . 421
A.4 Orderings of Monomials . . . . . . . . . . . . . . . . . . 424
A.5 Division in Polynomial Rings; Gr¨obner Bases . . . . . . . 427
A.6 Division in Power Series Rings; Standard Bases . . . . . 438
A.7 Division in the Ring of Germs . . . . . . . . . . . . . . . 444
B Modules 447
B.1 Submodules of Zn . . . . . . . . . . . . . . . . . . . . . . 447
B.2 Modules of Vector Fields . . . . . . . . . . . . . . . . . . 449
C Format 2b: Generated Recursive (Hori) 451
C.1 Format 2b, Linear Case (for Chapter 3) . . . . . . . . . . 451
C.2 Format 2b, Nonlinear Case (for Chapter 4) . . . . . . . . 457
D Format 2c: Generated Recursive (Deprit) 463
D.1 Format 2c, Linear Case (for Chapter 3) . . . . . . . . . . 463
D.2 Format 2c, Nonlinear Case (for Chapter 4) . . . . . . . . 471
E On Some Algorithms in Linear Algebra 477
References 481
Index 489
coordinates in a prescribed manner.
2. An unfolding of the normal form is obtained. This is intended to be

the simplest form into which all systems close to the original system
can be transformed. It will contain parameters, called unfolding
parameters, that are not present in the normal form found in step 1.