Author(s): Ian R. Porteous
Edition: 1
Publisher: Van Nostrand Reinhold Company
Year: 1969
Language: English
Pages: 472
Tags: Mathematics, Geometry and Topology
CONTENTS
FOREWORD 7
Acknowledgments; references and symbols
CHAPTER 0 GUIDE 1
CHAPTER 1 MAPS 4
Membership; maps; subsets and quotients; forwards and back-
wards; pairs; equivalences; products on a set; union and intersection;
natural numbers; products on ω;Σ and Π; order properties of ω
CHAPTER 2 REAL AND COMPLEX NUMBERS 26
Groups; rings; polynomials; ordered rings; absolute value; the
ring of integers; fields; the rational field; bounded subsets;
the >-> notation; the real field; convergence; the complex field;
the exponential maps
CHAPTER 3 LINEAR SPACES 53
Linear spaces; linear maps; linear sections; linear subspaces;
linear injections and surjections; linear products; linear spaces
of linear maps; bilinear maps; algebras; matrices; the algebras
⁸K; one-sided ideals; modules
CHAPTER 4 AFFINE SPACES 74
Affine spaces; translations; affine maps; affine subspaces; affine
subspaces of a linear space; lines in an affine space; convexity;
affine products; comment
CHAPTER 5 QUOTIENT STRUCTURES 85
Linear quotients; quotient groups; ideals; exact sequences;
diagram-chasing; the dual of an exact sequence; more diagram-chasing;
sections of a linear surjection; analogues for group maps; orbits
CHAPTER 6 FINITE-DIMENSIONAL SPACES 100
Linear dependence; the basis theorem; rank; matrices; finite-dimensional
algebras; minimal left ideals
CHAPTER 7 DETERMINANTS 116
Frames; elementary basic framings; permutations of n; the
determinant; transposition; determinants of endomorphisms;
the absolute determinant; applications; the sides of a hyperplane;
orientation
CHAPTER 8 DIRECT SUM 132
Direct sum; ²K-modules and maps; linear complements; complements
and quotients; spaces of linear complements; Grassmannians
CHAPTER 9 ORTHOGONAL SPACES 146
Real orthogonal spaces; invertible elements; linear correlations;
non-degenerate spaces; orthogonal maps; adjoints; examples of
adjoints; orthogonal annihilators; the basis theorem; reflections;
signature; Witt decompositions; neutral spaces; positive—definite
spaces; euclidean spaces; spheres; complex orthogonal spaces
CHAPTER 10 QUATERNIONS 174
The algebra H; automorphisms and anti-automorphisms of H;
rotations of R⁴; linear spaces over H; tensor product of algebras;
automorphisms and anti-automorphisms of ⁸K
CHAPTER 11 CORRELATIONS 198
Semi-linear maps; correlations; equivalent correlations; algebra
anti-involutions; correlated spaces; detailed classification
theorems; positive-definite spaces; particular adjoint anti-involutions;
groups of correlated automorphisms
CHAPTER 12 QUADRIC GRASSMANNIANS 223
Grassmannians; quadric Grassmannians; affine quadrics; real
affine quadrics; charts on quadric Grassmannians; Grassmannians
as coset spaces; quadric Grassmannians as coset spaces;
Cayley charts; Grassmannians as quadric Grassmannians;
further coset space representations
CHAPTER 13 CLIFFORD ALGEBRAS 240
Orthonormal subsets; the dimension of a Clifford algebra;
universal Clifford algebras; construction of the algebras;
complex Clifford algebras; involuted fields; involutions and
anti-involutions; the Clifford group; the uses of conjugation;
the map N; the Pfaffian chart; Spin groups; The Radon—Hurwitz
numbers
CHAPTER 14 THE CAYLEY ALGEBRA 277
Real division algebras; alternative division algebras; the Cayley
algebra; Hamilton triangles; Cayley triangles; further results;
the Cayley projective line and plane
CHAPTER 15 NORMED LINEAR SPACES 288
Norms; open and closed balls; open and closed sets; continuity;
complete normed affine spaces; equivalence of norms; the norm
of a continuous linear map; continuous bilinear maps; inversion
CHAPTER 16 TOPOLOGICAL SPACES 311
Topologies; continuity; subspaces and quotient spaces; closed
sets; limits; covers; compact spaces; Hausdorff spaces; open,
closed and compact maps; product topology; connectedness
CHAPTER 17 TOPOLOGICAL GROUPS AND MANIFOLDS 336
Topological groups; homogeneous spaces; topological manifolds;
Grassmannians; quadric Grassmannians; invariance of
domain
CHAPTER 18 AFFINE APPROXIMATION 353
Tangency; differentiable maps; complex differentiable maps;
properties of differentials; singularities of a map
CHAPTER 19 THE INVERSE FUNCTION THEOREM 375
The increment formula; the inverse function theorem; the
implicit function theorem; smooth subsets; local maxima and
minima; the rank theorem; the fundamental theorem of algebra;
higher differentials
CHAPTER 20 SMOOTH MANIFOLDS 399
Smooth manifolds and maps; submanifolds and products of
manifolds; dimension; tangent bundles and maps; particular
tangent spaces; smooth embeddings and projections; embeddings
of projective planes; tangent vector fields; Lie groups;
Lie algebras
BIBLIOGRAPHY 435
LIST OF SYMBOLS 439
INDEX 447
