Page 14 - Advanced Linear Algebra

xvi Contents

Quadratic Forms, 264
Orthogonality, 265
Linear Functionals, 268
Orthogonal Complements and Orthogonal Direct Sums, 269
Isometries, 271
Hyperbolic Spaces, 272
Nonsingular Completions of a Subspace, 273
The Witt Theorems: A Preview, 275
The Classification Problem for Metric Vector Spaces, 276
Symplectic Geometry, 277
The Structure of Orthogonal Geometries: Orthogonal Bases, 282
The Classification of Orthogonal Geometries:
Canonical Forms, 285
The Orthogonal Group, 291
The Witt Theorems for Orthogonal Geometries, 294
Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 295
Exercises, 297
12 Metric Spaces, 301
The Definition, 301
Open and Closed Sets, 304
Convergence in a Metric Space, 305
The Closure of a Set, 306
Dense Subsets, 308
Continuity, 310
Completeness, 311
Isometries, 315
The Completion of a Metric Space, 316
Exercises, 321
13 Hilbert Spaces, 325
A Brief Review, 325
Hilbert Spaces, 326
Infinite Series, 330
An Approximation Problem, 331
Hilbert Bases, 335
Fourier Expansions, 336
A Characterization of Hilbert Bases, 346
Hilbert Dimension, 346
A Characterization of Hilbert Spaces, 347
The Riesz Representation Theorem, 349
Exercises, 352
14 Tensor Products, 355
Universality, 355
Bilinear Maps, 359
Tensor Products, 361