Banach-Hilbert spaces, vector measures, and group representations

An elementary introduction to classical analysis on normed spaces, with special attention paid to fixed points, calculus, and ordinary differential equations. It contains a full treatment of vector measures on delta rings without assuming any scalar measure theory and hence should fit well into existing courses. The relation between group representations and almost periodic functions is presented. The mean values offer an infinite-dimensional analogue of measure theory on finite-dimensional Euclidean spaces. The work should be suitable for beginners who want to get through the basic material as soon as possible and then do their own research immediately. Front Cover......Page 1 Title......Page 4 Copyright Information......Page 5 Preface......Page 6 Contents ......Page 10 Introduction......Page 16 1-1 Standard Finite Dimensional Vector Spaces......Page 26 1-2 Convergent Sequences in Metric Spaces......Page 28 1-3 Continuous Maps......Page 29 1-4 Open Sets......Page 32 1-5 Closures of Sets......Page 33 1-6 Characterization of Continuity ......Page 35 1-7 Duality of Closure-Interior Operators ......Page 37 1-8 Partition of Unity ......Page 39 2-1 Cauchy Sequences......Page 42 2-2 Bounded Sets ......Page 43 2-3 Upper and Lower Limits ......Page 44 2-4 Complete Sets ......Page 46 2-5 Precompact Sets ......Page 48 2-6 Compactness ......Page 51 2-7 Continuous Maps on Compact Spaces ......Page 54 2-8 Uniform Continuity ......Page 56 2-9 Connected Sets ......Page 58 2-10. Components ......Page 61 3-1 Uniform Convergence......Page 63 3-2 Bounded Continuous Functions ......Page 64 3-3 Sequence Spaces ......Page 67 3-4 Continuous Linear Maps ......Page 70 3-5 Examples of Continuous Linear Maps ......Page 74 3-6 Finite Dimensional Normed Spaces ......Page 76 3-7 Infinite Dimensional Compact Sets ......Page 79 3-8 Approximation in Function Spaces ......Page 82 4-1 Geometrically Independent Sets......Page 86 4-2 Convex Sets in Normed Spaces ......Page 90 4-3 Simplexes ......Page 92 4-4 Affine Maps ......Page 93 4-5 Simplicial Complexes ......Page 95 4-6 Small Simplexes ......Page 98 4-7 Barycentric Subdivisions ......Page 100 4-8 Simplicial Approximations ......Page 102 4-9 Existence of Simplicial Approximations ......Page 104 4-10. A Combinatorial Lemma with Application ......Page 106 5-1 Antipodal Maps......Page 110 5-2 Retracts and Fixed Points ......Page 113 5-3 Fixed Points of Compact Maps......Page 116 5-4 Compact Fields and their Homotopies......Page 117 5-5 Extension Property......Page 120 5-6 Properties of Compact Fields in Normed Spaces......Page 124 6-1 Transfmite Induction......Page 129 6-2 Hahn-Banach Extension Theorems......Page 131 6-3 Extension of Continuous Linear Forms......Page 133 6-4 Closed Hyperplanes......Page 135 6-5 Separation by Hyperplanes......Page 138 6-6 Extreme Points......Page 140 6-7 Baire's Property......Page 142 6-8 Uniform Boundedness......Page 143 6-9 Open Map and Closed Graph Theorems......Page 145 7-1 Bidual Spaces......Page 149 7-2 Quotient Spaces......Page 151 7-3 Duality of Subspaces and Quotients......Page 153 7-4 Direct Sums......Page 155 7-5 Transposes......Page 159 7-6 Reflexive Spaces......Page 162 7-7 Weak convergence......Page 164 7-8 Weak-Star Convergence......Page 167 8-1 Derivatives of Vector Maps......Page 169 8-2 Integrals of Regulated Maps......Page 170 8-3 Fundamental Theorems of Calculus......Page 173 8-4 Holomorpbic Maps of One Complex Variable......Page 176 8-5 Series Expansion......Page 180 8-6 Spectrum......Page 185 8-7 Spectral Radius......Page 189 8-8 Holomorphic Maps of an Operator......Page 191 9-1 Differentiable Maps......Page 197 9-2 Mean-Value Theorem......Page 200 9-3 Partial Derivatives......Page 203 9-4 Fixed Points of Contractions......Page 207 9-5 Inverse and Implicit Mapping Theorems......Page 208 9-6 Local Properties of Differentiable Maps......Page 212 10-1 Multilinear Maps on Banach Spaces......Page 216 10-2 Polynomials on Banach Spaces ......Page 219 10-3 Higher Derivatives ......Page 223 10-4 Cn-Maps ......Page 226 10-5 Taylor's Expansion ......Page 230 10-6 Higher Chain Formula and Higher Product Formula ......Page 234 11-1 Local Existence and Uniqueness......Page 238 11-2 Integral Curves ......Page 241 11-3 Linear Equations ......Page 243 11-4 Exponential Functions of Matrices ......Page 248 11-5 Global Dependence on Initial Conditions ......Page 250 11-6 Solutions without Uniqueness ......Page 257 12-1 Basic Properties......Page 260 12-2 Riesz-Schauder Theory ......Page 264 12-3 Spectrum of a Compact Operator ......Page 268 12-4 Existence of Invariant Subspaces ......Page 269 12-5 Fredholm Operators ......Page 271 13-1 Complex Inner Product Spaces......Page 276 13-2 Orthogonality in Inner Product Spaces ......Page 278 13-3 Orthonormal Bases of Hubert Spaces ......Page 280 13-4 Orthogonal Complements ......Page 283 13-5 Adjoints ......Page 285 13-6 Quadratic Forms ......Page 289 13-7 Normal Operators ......Page 291 13-8 Self-Adjoint Operators ......Page 293 13-9 Projectors and Closed Vector Subspaces ......Page 295 13-10 Partial Order of Operators ......Page 299 13-11 Eigenvalues ......Page 303 14-1 Spectrum of an Operator......Page 306 14-2 Approximate Spectrum ......Page 307 14-3 Weak Convergence ......Page 310 14-4 Diagonal Operators ......Page 312 14-5 Compact Operators ......Page 314 14-6 Functional Calculus of Self-Adjoint Operators ......Page 320 14-7 Polar Decomposition ......Page 325 15-1 Algebraic Tensor Products of Vector Spaces......Page 328 15-2 Tensor Products of Linear Maps ......Page 330 15-3 Independent Sets in Tensor Products ......Page 332 15-4 Matrix Representations ......Page 334 15-5 Projective Norms on Tensor Products ......Page 338 15-6 Inductive Norms ......Page 342 15-7 Tensor Product of Hilbert Spaces ......Page 344 16-1 Ordered Vector Spaces......Page 350 16-2 Lattice Structure ......Page 351 16-3 Decomposition Property ......Page 354 16-4 Extension of Positive Linear Forms ......Page 356 16-5 Order Bounded Linear Forms ......Page 358 17-1 Semirings......Page 361 17-2 Charges and Associated Integrals ......Page 362 17-3 Finite Variation ......Page 365 17-4 Absolutely Convergent Charges ......Page 367 17-5 Countable Additivity on Rings ......Page 370 17-6 Vector Measures ......Page 373 17-7 Lebesgue-Stieltjes Measures ......Page 375 18-1 Uniqueness of Extension......Page 379 18-2 Outer Measures ......Page 381 18-3 Extension to Decent Sets ......Page 385 19-1 Measurable Sets......Page 387 19-2 Measurable Functions ......Page 389 19-3 Limits of Measurable Functions ......Page 392 19-4 Approximations by Simple Functions ......Page 393 19-5 Measurable Maps ......Page 395 19-6 More Properties ......Page 397 20-1 Upper Functions......Page 401 20-2 Almost Everywhere ......Page 404 20-3 Seeds of the Theory ......Page 406 20-4 Sigma Finiteness ......Page 407 20-5 Comparison of Two Positive Measures ......Page 409 21-1 Extension to Integrable Sets......Page 412 21-2 Integrals of Vector Maps ......Page 414 21-3 Lp-Spaces for 1