Finite-Dimensional Spaces

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.