A. Audience. This treatise (consisting of the present VoU and of VoUI, to be published) is primarily intended to be a textbook for a core course in mathematics at the advanced undergraduate or the beginning graduate level. The treatise should also be useful as a textbook for selected stu dents in honors programs at the sophomore and junior level. Finally, it should be of use to theoretically inclined scientists and engineers who wish to gain a better understanding of those parts of mathemat ics that are most likely to help them gain insight into the conceptual foundations of the scientific discipline of their interest. B. Prerequisites. Before studying this treatise, a student should be familiar with the material summarized in Chapters 0 and 1 of Vol.1. Three one-semester courses in serious mathematics should be sufficient to gain such fa miliarity. The first should be an introduction to contemporary math ematics and should cover sets, families, mappings, relations, number systems, and basic algebraic structures. The second should be an in troduction to rigorous real analysis, dealing with real numbers and real sequences, and with limits, continuity, differentiation, and integration of real functions of one real variable. The third should be an intro duction to linear algebra, with emphasis on concepts rather than on computational procedures. C. Organization.
Inhaltsverzeichnis
0 Basic Mathematics.
00 Notations.
01 Sets, Partitions.
02 Families, Lists, Matrices.
03 Mappings.
04 Families of Sets; Families and Sets of Mappings.
05 Finite Sets.
06 Basic Algebra.
07 Summations.
08 Real Analysis.
1 Linear Spaces.
11 Basic Definitions.
12 Subspaces.
13 Linear Mappings.
14 Spaces of Mappings, Product Spaces.
15 Linear Combinations, Linear Independence, Bases.
16 Matrices, Elimination of Unknowns.
17 Dimension.
18 Lineons.
19 Projections, Idempotents.
Problems.
2 Duality, Bilinearity.
21 Dual Spaces, Transposition, Annihilators.
22 The Second Dual Space.
23 Dual Bases.
24 Bilinear Mappings.
25 Tensor Products.
26 The Trace.
27 Bilinear Forms and Quadratic Forms.
Problems.
3 Flat Spaces.
31 Actions of Groups.
32 Flat Spaces and Flats.
33 Flat Mappings.
34 Charge Distributions, Barycenters, Mass-Points.
35 Flat Combinations.
36 Flat Functions.
37 Convex Sets.
38 Half-Spaces.
Problems.
4 Inner-Product Spaces, Euclidean Spaces.
41 Inner-Product Spaces.
42 Genuine Inner-Product Spaces.
43 Orthogonal Mappings.
44 Induced Inner Products.
45 Euclidean Spaces.
46 Genuine Euclidean Spaces, Congruences.
47 Double-Signed Inner-Product Spaces.
Problems.
5 Topology.
51 Cells and Norms.
52 Bounded Sets, Operator Norms.
53 Neighborhoods, Open and Closed Sets.
54 Topology of Convex Sets.
55 Sequences.
56 Continuity, Uniform Continuity.
57 Limits.
58 Compactness.
Problems.
6 Differential Calculus.
61 Differentiation of Processes.
62 Small and Confined Mappings.
63 Gradients, Chain Rule.
64 Constricted Mappings.
65 Partial Gradients, Directional Derivatives.
66 The General Product Rule.
67 Divergence, Laplacian.
68 Local Inversion, Implicit Mappings.
69 Extreme Values, Constraints.
610 Integral Representations.
611 Curl, Symmetry of Second Gradients.
612 Lineonic Exponentials.
Problems.
7 Coordinate Systems.
71 Coordinates in Flat Spaces.
72 Connection Components, Components of Gradients.
73 Coordinates in Euclidean Spaces.
74 Special Coordinate Systems.
Problems.
8 Spectral Theory.
81 Disjunct Families, Decompositions.
82 Spectral Values and Spectral Spaces.
83 Orthogonal Families of Subspaces.
84 The Structure of Symmetric Lineons.
85 Lineonic Extensions, Lineonic Square Roots and Logarithms.
86 Polar Decomposition.
87 The Structure of Skew Lineons.
88 The Structure of Normal and of Orthogonal Lineons.
89 Complex Spaces, Unitary Spaces.
810 Complex Spectra.
Problems.
9 The Structure of General Lineons.
91 Elementary Decompositions.
92 Lineonic Polynomial Functions.
93 The Structure of Elementary Lineons.
94 Canonical Matrices.
95 Similarity, Elementary Divisors.
Problems.
of Theorem Titles.
of Special Notations.
of Multiple-Letter Symbols.
