Banach-Hilbert spaces, vector measures, and group representations

Description

Provides an elementary introduction to classical analysis on normed spaces, with special attention paid to fixed points, calculus, and ordinary differential equations. Ideal for beginners who want to get through the basic material as soon as possible and then do their own research immediately. Title......Page 1 Preface......Page 3 Contents ......Page 7 Introduction......Page 13 1-1 Standard Finite Dimensional Vector Spaces......Page 23 1-2 Convergent Sequences in Metric Spaces......Page 25 1-3 Continuous Maps......Page 26 1-4 Open Sets......Page 29 1-5 Closures of Sets......Page 30 1-6 Characterization of Continuity ......Page 32 1-7 Duality of Closure-Interior Operators ......Page 34 1-8 Partition of Unity ......Page 36 2-1 Cauchy Sequences ......Page 39 2-2 Bounded Sets ......Page 40 2-3 Upper and Lower Limits ......Page 41 2-4 Complete Sets ......Page 43 2-5 Precompact Sets ......Page 45 2-6 Compactness ......Page 48 2-7 Continuous Maps on Compact Spaces ......Page 51 2-8 Uniform Continuity ......Page 53 2-9 Connected Sets ......Page 55 2-10. Components ......Page 58 3-1 Uniform Convergence ......Page 60 3-2 Bounded Continuous Functions ......Page 61 3-3 Sequence Spaces ......Page 64 3-4 Continuous Linear Maps ......Page 67 3-5 Examples of Continuous Linear Maps ......Page 71 3-6 Finite Dimensional Normed Spaces ......Page 73 3-7 Infinite Dimensional Compact Sets ......Page 76 3-8 Approximation in Function Spaces ......Page 79 4-1 Geometrically Independent Sets ......Page 83 4-2 Convex Sets in Normed Spaces ......Page 87 4-3 Simplexes ......Page 89 4-4 Affine Maps ......Page 90 4-5 Simplicial Complexes ......Page 92 4-6 Small Simplexes ......Page 95 4-7 Barycentric Subdivisions ......Page 97 4-8 Simplicial Approximations ......Page 99 4-9 Existence of Simplicial Approximations ......Page 101 4-10. A Combinatorial Lemma with Application ......Page 103 5-1 Antipodal Maps ......Page 107 5-2 Retracts and Fixed Points ......Page 110 5-3 Fixed Points of Compact Maps......Page 113 5-4 Compact Fields and their Homotopies......Page 114 5-5 Extension Property......Page 117 5-6 Properties of Compact Fields in Normed Spaces......Page 121 6-1 Transfmite Induction......Page 126 6-2 Hahn-Banach Extension Theorems......Page 128 6-3 Extension of Continuous Linear Forms......Page 130 6-4 Closed Hyperplanes......Page 132 6-5 Separation by Hyperplanes......Page 135 6-6 Extreme Points......Page 137 6-7 Baire's Property......Page 139 6-8 Uniform Boundedness......Page 140 6-9 Open Map and Closed Graph Theorems......Page 142 7-1 Bidual Spaces......Page 146 7-2 Quotient Spaces......Page 148 7-3 Duality of Subspaces and Quotients......Page 150 7-4 Direct Sums......Page 152 7-5 Transposes......Page 156 7-6 Reflexive Spaces......Page 159 7-7 Weak convergence......Page 161 7-8 Weak-Star Convergence......Page 164 8-1 Derivatives of Vector Maps......Page 166 8-2 Integrals of Regulated Maps......Page 167 8-3 Fundamental Theorems of Calculus......Page 170 8-4 Holomorpbic Maps of One Complex Variable......Page 173 8-5 Series Expansion......Page 177 8-6 Spectrum......Page 182 8-7 Spectral Radius......Page 186 8-8 Holomorphic Maps of an Operator......Page 188 9-1 Differentiable Maps......Page 194 9-2 Mean-Value Theorem......Page 197 9-3 Partial Derivatives......Page 200 9-4 Fixed Points of Contractions......Page 204 9-5 Inverse and Implicit Mapping Theorems......Page 205 9-6 Local Properties of Differentiable Maps......Page 209 10-1 Multilinear Maps on Banach Spaces ......Page 213 10-2 Polynomials on Banach Spaces ......Page 216 10-3 Higher Derivatives ......Page 220 10-4 Cn-Maps ......Page 223 10-5 Taylor's Expansion ......Page 227 10-6 Higher Chain Formula and Higher Product Formula ......Page 231 11-1 Local Existence and Uniqueness ......Page 235 11-2 Integral Curves ......Page 238 11-3 Linear Equations ......Page 240 11-4 Exponential Functions of Matrices ......Page 245 11-5 Global Dependence on Initial Conditions ......Page 247 11-6 Solutions without Uniqueness ......Page 254 12-1 Basic Properties ......Page 257 12-2 Riesz-Schauder Theory ......Page 261 12-3 Spectrum of a Compact Operator ......Page 265 12-4 Existence of Invariant Subspaces ......Page 266 12-5 Fredholm Operators ......Page 268 13-1 Complex Inner Product Spaces ......Page 273 13-2 Orthogonality in Inner Product Spaces ......Page 275 13-3 Orthonormal Bases of Hubert Spaces ......Page 277 13-4 Orthogonal Complements ......Page 280 13-5 Adjoints ......Page 282 13-6 Quadratic Forms ......Page 286 13-7 Normal Operators ......Page 288 13-8 Self-Adjoint Operators ......Page 290 13-9 Projectors and Closed Vector Subspaces ......Page 292 13-10 Partial Order of Operators ......Page 296 13-11 Eigenvalues ......Page 300 14-1 Spectrum of an Operator ......Page 303 14-2 Approximate Spectrum ......Page 304 14-3 Weak Convergence ......Page 307 14-4 Diagonal Operators ......Page 309 14-5 Compact Operators ......Page 311 14-6 Functional Calculus of Self-Adjoint Operators ......Page 317 14-7 Polar Decomposition ......Page 322 15-1 Algebraic Tensor Products of Vector Spaces ......Page 325 15-2 Tensor Products of Linear Maps ......Page 327 15-3 Independent Sets in Tensor Products ......Page 329 15-4 Matrix Representations ......Page 331 15-5 Projective Norms on Tensor Products ......Page 335 15-6 Inductive Norms ......Page 339 15-7 Tensor Product of Hilbert Spaces ......Page 341 16-1 Ordered Vector Spaces ......Page 347 16-2 Lattice Structure ......Page 348 16-3 Decomposition Property ......Page 351 16-4 Extension of Positive Linear Forms ......Page 353 16-5 Order Bounded Linear Forms ......Page 355 17-1 Semirings ......Page 358 17-2 Charges and Associated Integrals ......Page 359 17-3 Finite Variation ......Page 362 17-4 Absolutely Convergent Charges ......Page 364 17-5 Countable Additivity on Rings ......Page 367 17-6 Vector Measures ......Page 370 17-7 Lebesgue-Stieltjes Measures ......Page 372 18-1 Uniqueness of Extension ......Page 376 18-2 Outer Measures ......Page 378 18-3 Extension to Decent Sets ......Page 382 19-1 Measurable Sets ......Page 384 19-2 Measurable Functions ......Page 386 19-3 Limits of Measurable Functions ......Page 389 19-4 Approximations by Simple Functions ......Page 390 19-5 Measurable Maps ......Page 392 19-6 More Properties ......Page 394 20-1 Upper Functions ......Page 398 20-2 Almost Everywhere ......Page 401 20-3 Seeds of the Theory ......Page 403 20-4 Sigma Finiteness ......Page 404 20-5 Comparison of Two Positive Measures ......Page 406 21-1 Extension to Integrable Sets ......Page 409 21-2 Integrals of Vector Maps ......Page 411 21-3 Lp-Spaces for 1