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Algebra, Topology, Differential Calculus and Optimization for CS 2017.pdf
Cover
Contents
Chapter 1 Introduction
Chapter 2 Vector Spaces, Bases, Linear Maps
Chapter 3 Matrices and Linear Maps
Chapter 4 Direct Sums, The Dual Space, Duality
Chapter 5 Determinants
Chapter 6 Gaussian Elimination,LU-Factorization, Cholesky Factorization, Reduced Row Echelon Form
Chapter 7 Vector Norms and Matrix Norms
Chapter 8 Eigenvectors and Eigenvalues
Chapter 9 Iterative Methods for Solving Linear Systems
Chapter 10 Euclidean Spaces
Chapter 11 QR-Decomposition for Arbitrary Matrices
Chapter 12 Basics of Affine Geometry
Chapter 13 Embedding an Affine Space in a Vector Space
Chapter 14 Basics of Projective Geometry
Chapter 15 The Cartan{Dieudonn´e Theorem
Chapter 16 Hermitian Spaces
Chapter 17 Isometries of Hermitian Spaces
Chapter 18 Spectral Theorems in Euclidean and Hermitian Spaces
Chapter 19 Variational Approximation of Boundary-Value Problems;Introduction to the Finite Elements Method
Chapter 20 Singular Value Decomposition and Polar Form
Chapter 21 Applications of SVD and Pseudo-Inverses
Chapter 22 The Geometry of Bilinear Forms;Witt’s Theorem; The Cartan{Dieudonn´e Theorem
Chapter 23 Polynomials, Ideals and PID’s
Chapter 24 Annihilating Polynomials and the Primary Decomposition
Chapter 25 UFD’s, Noetherian Rings, Hilbert’s Basis Theorem
Chapter 26 Tensor Algebras and Symmetric Algebras
Chapter 27 Exterior Tensor Powers and Exterior Algebras
Chapter 28 Introduction to Modules; Modules over a PID
Chapter 29 The Rational Canonical Form and Other Normal Forms
Chapter 30 Topology
Chapter 31 A Detour On Fractals
Chapter 32 Differential Calculus
Chapter 33 Quadratic Optimization Problems
Chapter 34 Schur Complements and Applications
Chapter 35 Convex Sets, Cones, H-Polyhedra
Chapter 36 Linear Programs
Chapter 37 The Simplex Algorithm
Chapter 38 Linear Programming and Duality
Chapter 39 Extrema of Real-Valued Functions
Chapter 40 Newton’s Method and Its Generalizations
Chapter 41 Basics of Hilbert Spaces
Chapter 42 General Results of Optimization Theory
Chapter 43 Introduction to Nonlinear Optimization
Chapter 44 Soft Margin Support Vector Machines
Chapter 45 Total Orthogonal Families in Hilbert Spaces
Chapter 46 Appendix: Zorn’s Lemma; Some Applications
Bibliography
Algebra, Topology, Differential Calculus, and
Optimization Theory
For Computer Science and Engineering
Jean Gallier and Jocelyn Quaintance
Department of Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104, USA
e-mail: jean@cis.upenn.edu
c Jean Gallier
April 24, 2017
2
Contents
1 Introduction
2 Vector Spaces, Bases, Linear Maps
Linear Independence, Subspaces
2.1 Groups, Rings, and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
. . . . . . . . . . . . . . . . . . . . . . . .
2.4 Bases of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
2.6 Quotient Spaces
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
Linear Maps
3 Matrices and Linear Maps
3.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
3.2 Haar Basis Vectors and a Glimpse at Wavelets
3.3 The Effect of a Change of Bases on Matrices
. . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
13
15
15
26
33
38
45
52
53
55
55
71
88
91
4 Direct Sums, The Dual Space, Duality
Sums, Direct Sums, Direct Products . . . . . . . . . . . . . . . . . . . . . .
93
4.1
93
4.2 The Dual Space E∗ and Linear Forms
. . . . . . . . . . . . . . . . . . . . . 108
4.3 Hyperplanes and Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.4 Transpose of a Linear Map and of a Matrix . . . . . . . . . . . . . . . . . . 127
4.5 The Four Fundamental Subspaces
. . . . . . . . . . . . . . . . . . . . . . . 136
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.6
5 Determinants
141
5.1 Permutations, Signature of a Permutation . . . . . . . . . . . . . . . . . . . 141
5.2 Alternating Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.3 Definition of a Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.4
Inverse Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . . 155
5.5
Systems of Linear Equations and Determinants . . . . . . . . . . . . . . . . 158
5.6 Determinant of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.7 The Cayley–Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.8 Permanents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3
4
CONTENTS
5.9
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6 Gaussian Elimination, LU, Cholesky, Echelon Form
169
6.1 Motivating Example: Curve Interpolation . . . . . . . . . . . . . . . . . . . 169
6.2 Gaussian Elimination and LU -Factorization . . . . . . . . . . . . . . . . . . 173
6.3 Gaussian Elimination of Tridiagonal Matrices . . . . . . . . . . . . . . . . . 199
6.4
SPD Matrices and the Cholesky Decomposition . . . . . . . . . . . . . . . . 202
6.5 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.6 Transvections and Dilatations . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7 Vector Norms and Matrix Norms
231
7.1 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.2 Matrix Norms
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.3 Condition Numbers of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.4 An Application of Norms: Inconsistent Linear Systems . . . . . . . . . . . . 259
7.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
8 Eigenvectors and Eigenvalues
263
8.1 Eigenvectors and Eigenvalues of a Linear Map . . . . . . . . . . . . . . . . . 263
8.2 Reduction to Upper Triangular Form . . . . . . . . . . . . . . . . . . . . . . 270
8.3
Location of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
8.4
9 Iterative Methods for Solving Linear Systems
279
9.1 Convergence of Sequences of Vectors and Matrices
. . . . . . . . . . . . . . 279
9.2 Convergence of Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . 282
9.3 Methods of Jacobi, Gauss-Seidel, and Relaxation . . . . . . . . . . . . . . . 284
9.4 Convergence of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
9.5
10 Euclidean Spaces
297
10.1 Inner Products, Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . 297
10.2 Orthogonality, Duality, Adjoint of a Linear Map . . . . . . . . . . . . . . . 305
10.3 Linear Isometries (Orthogonal Transformations) . . . . . . . . . . . . . . . . 317
10.4 The Orthogonal Group, Orthogonal Matrices
. . . . . . . . . . . . . . . . . 320
10.5 QR-Decomposition for Invertible Matrices . . . . . . . . . . . . . . . . . . . 322
10.6 Some Applications of Euclidean Geometry . . . . . . . . . . . . . . . . . . . 326
10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
11 QR-Decomposition for Arbitrary Matrices
329
11.1 Orthogonal Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
11.2 QR-Decomposition Using Householder Matrices . . . . . . . . . . . . . . . . 333
CONTENTS
5
11.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
12 Basics of Affine Geometry
339
12.1 Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
12.2 Examples of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
12.3 Chasles’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
12.4 Affine Combinations, Barycenters . . . . . . . . . . . . . . . . . . . . . . . . 349
12.5 Affine Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.6 Affine Independence and Affine Frames . . . . . . . . . . . . . . . . . . . . . 358
12.7 Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
12.8 Affine Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
12.9 Affine Geometry: A Glimpse
. . . . . . . . . . . . . . . . . . . . . . . . . . 372
12.10 Affine Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
12.11 Intersection of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
13 Embedding an Affine Space in a Vector Space
381
13.1 The “Hat Construction,” or Homogenizing . . . . . . . . . . . . . . . . . . . 381
13.2 Affine Frames of E and Bases of ˆE . . . . . . . . . . . . . . . . . . . . . . . 388
13.3 Another Construction of ˆE . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
13.4 Extending Affine Maps to Linear Maps . . . . . . . . . . . . . . . . . . . . . 393
14 Basics of Projective Geometry
399
14.1 Why Projective Spaces? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
14.2 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
14.3 Projective Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
14.4 Projective Frames
14.5 Projective Maps
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
14.6 Finding a Homography Between Two Projective Frames . . . . . . . . . . . 432
14.7 Affine Patches
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
14.8 Projective Completion of an Affine Space
. . . . . . . . . . . . . . . . . . . 448
14.9 Making Good Use of Hyperplanes at Infinity . . . . . . . . . . . . . . . . . 453
14.10 The Cross-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
14.11 Fixed Points of Homographies and Homologies
. . . . . . . . . . . . . . . . 460
14.12 Duality in Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 474
14.13 Cross-Ratios of Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
14.14 Complexification of a Real Projective Space . . . . . . . . . . . . . . . . . . 480
14.15 Similarity Structures on a Projective Space . . . . . . . . . . . . . . . . . . 482
14.16 Some Applications of Projective Geometry . . . . . . . . . . . . . . . . . . . 491
15 The Cartan–Dieudonn´e Theorem
497
15.1 The Cartan–Dieudonn´e Theorem for Linear Isometries . . . . . . . . . . . . 497
15.2 Affine Isometries (Rigid Motions) . . . . . . . . . . . . . . . . . . . . . . . . 509
15.3 Fixed Points of Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
6
CONTENTS
15.4 Affine Isometries and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . 513
. . . . . . . . . . . . 519
15.5 The Cartan–Dieudonn´e Theorem for Affine Isometries
16 Hermitian Spaces
523
16.1 Hermitian Spaces, Pre-Hilbert Spaces
. . . . . . . . . . . . . . . . . . . . . 523
16.2 Orthogonality, Duality, Adjoint of a Linear Map . . . . . . . . . . . . . . . 532
16.3 Linear Isometries (Also Called Unitary Transformations) . . . . . . . . . . . 537
16.4 The Unitary Group, Unitary Matrices
. . . . . . . . . . . . . . . . . . . . . 539
16.5 Orthogonal Projections and Involutions
. . . . . . . . . . . . . . . . . . . . 542
16.6 Dual Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
16.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
17 Isometries of Hermitian Spaces
551
17.1 The Cartan–Dieudonn´e Theorem, Hermitian Case . . . . . . . . . . . . . . . 551
17.2 Affine Isometries (Rigid Motions) . . . . . . . . . . . . . . . . . . . . . . . . 560
18 Spectral Theorems
565
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
18.2 Normal Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
18.3 Self-Adjoint and Other Special Linear Maps . . . . . . . . . . . . . . . . . . 574
18.4 Normal and Other Special Matrices . . . . . . . . . . . . . . . . . . . . . . . 581
18.5 Conditioning of Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . 584
18.6 Rayleigh Ratios and the Courant-Fischer Theorem . . . . . . . . . . . . . . 587
18.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
19 Introduction to The Finite Elements Method
597
19.1 A One-Dimensional Problem: Bending of a Beam . . . . . . . . . . . . . . . 597
19.2 A Two-Dimensional Problem: An Elastic Membrane . . . . . . . . . . . . . 607
19.3 Time-Dependent Boundary Problems . . . . . . . . . . . . . . . . . . . . . . 610
20 Singular Value Decomposition and Polar Form
619
. . . . . . . . . . . . . . 619
20.1 Singular Value Decomposition for Square Matrices
. . . . . . . . . . . 627
20.2 Singular Value Decomposition for Rectangular Matrices
20.3 Ky Fan Norms and Schatten Norms
. . . . . . . . . . . . . . . . . . . . . . 630
20.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
21 Applications of SVD and Pseudo-Inverses
633
21.1 Least Squares Problems and the Pseudo-Inverse . . . . . . . . . . . . . . . . 633
21.2 Properties of the Pseudo-Inverse . . . . . . . . . . . . . . . . . . . . . . . . 638
21.3 Data Compression and SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
21.4 Principal Components Analysis (PCA) . . . . . . . . . . . . . . . . . . . . . 644
21.5 Best Affine Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
21.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
CONTENTS
7
22 The Geometry of Bilinear Forms; Witt’s Theorem
657
22.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
22.2 Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
22.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
22.4 Adjoint of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
22.5 Isometries Associated with Sesquilinear Forms . . . . . . . . . . . . . . . . . 676
22.6 Totally Isotropic Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
22.7 Witt Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
22.8 Symplectic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
22.9 Orthogonal Groups and the Cartan–Dieudonn´e Theorem . . . . . . . . . . . 698
22.10 Witt’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
23 Polynomials, Ideals and PID’s
711
23.1 Multisets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
23.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
23.3 Euclidean Division of Polynomials
. . . . . . . . . . . . . . . . . . . . . . . 718
23.4 Ideals, PID’s, and Greatest Common Divisors . . . . . . . . . . . . . . . . . 720
23.5 Factorization and Irreducible Factors in K[X] . . . . . . . . . . . . . . . . . 728
23.6 Roots of Polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
23.7 Polynomial Interpolation (Lagrange, Newton, Hermite) . . . . . . . . . . . . 739
24 Annihilating Polynomials; Primary Decomposition
747
. . . . . . . . . . . . 747
24.1 Annihilating Polynomials and the Minimal Polynomial
24.2 Minimal Polynomials of Diagonalizable Linear Maps
. . . . . . . . . . . . . 749
24.3 The Primary Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . 755
24.4 Nilpotent Linear Maps and Jordan Form . . . . . . . . . . . . . . . . . . . . 764
25 UFD’s, Noetherian Rings, Hilbert’s Basis Theorem
771
25.1 Unique Factorization Domains (Factorial Rings) . . . . . . . . . . . . . . . . 771
25.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . 785
25.3 Noetherian Rings and Hilbert’s Basis Theorem . . . . . . . . . . . . . . . . 791
25.4 Futher Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795
26 Tensor Algebras and Symmetric Algebras
797
26.1 Linear Algebra Preliminaries: Dual Spaces and Pairings
. . . . . . . . . . . 798
26.2 Tensors Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
26.3 Bases of Tensor Products
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 814
26.4 Some Useful Isomorphisms for Tensor Products . . . . . . . . . . . . . . . . 816
26.5 Duality for Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 820
26.6 Tensor Algebras
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
26.7 Symmetric Tensor Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830
26.8 Bases of Symmetric Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 835
26.9 Some Useful Isomorphisms for Symmetric Powers . . . . . . . . . . . . . . . 838
8
CONTENTS
26.10 Duality for Symmetric Powers . . . . . . . . . . . . . . . . . . . . . . . . . . 838
26.11 Symmetric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841
27 Exterior Tensor Powers and Exterior Algebras
845
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845
27.1 Exterior Tensor Powers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 850
27.2 Bases of Exterior Powers
. . . . . . . . . . . . . . . . 853
27.3 Some Useful Isomorphisms for Exterior Powers
27.4 Duality for Exterior Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 853
27.5 Exterior Algebras
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856
27.6 The Hodge ∗-Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860
27.7 Left and Right Hooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
27.8 Testing Decomposability . . . . . . . . . . . . . . . . . . . . . . . . . . . 872
27.9 The Grassmann-Pl¨ucker’s Equations and Grassmannians . . . . . . . . . 875
27.10 Vector-Valued Alternating Forms . . . . . . . . . . . . . . . . . . . . . . . . 879
28 Introduction to Modules; Modules over a PID
883
28.1 Modules over a Commutative Ring . . . . . . . . . . . . . . . . . . . . . . . 883
28.2 Finite Presentations of Modules . . . . . . . . . . . . . . . . . . . . . . . . . 892
28.3 Tensor Products of Modules over a Commutative Ring . . . . . . . . . . . . 898
28.4 Torsion Modules over a PID; Primary Decomposition . . . . . . . . . . . . . 901
28.5 Finitely Generated Modules over a PID . . . . . . . . . . . . . . . . . . . . 907
. . . . . . . . . . . . . . . . . . . . . . . . 923
28.6 Extension of the Ring of Scalars
29 Normal Forms; The Rational Canonical Form
929
29.1 The Torsion Module Associated With An Endomorphism . . . . . . . . . . 929
29.2 The Rational Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . 937
29.3 The Rational Canonical Form, Second Version . . . . . . . . . . . . . . . . . 944
29.4 The Jordan Form Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 945
29.5 The Smith Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948
30 Topology
961
30.1 Metric Spaces and Normed Vector Spaces . . . . . . . . . . . . . . . . . . . 961
30.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967
30.3 Continuous Functions, Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 976
30.4 Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983
30.5 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992
30.6 Sequential Compactness in Metric Spaces
. . . . . . . . . . . . . . . . . . . 1003
30.7 Complete Metric Spaces and Compactness . . . . . . . . . . . . . . . . . . . 1011
30.8 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . 1012
30.9 Continuous Linear and Multilinear Maps . . . . . . . . . . . . . . . . . . . . 1017
30.10 Normed Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
30.11 Futher Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
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