Banach, Fréchet, Hilbert and Neumann Spaces

This book is the first of a set dedicated to the mathematical tools used in partial differential equations derived from physics. Its focus is on normed or semi-normed vector spaces, including the spaces of Banach, Fréchet and Hilbert, with new developments on Neumann spaces, but also on extractable spaces. The author presents the main properties of these spaces, which are useful for the construction of Lebesgue and Sobolev distributions with real or vector values and for solving partial differential equations. Differential calculus is also extended to semi-normed spaces. Simple methods, semi-norms, sequential properties and others are discussed, making these tools accessible to the greatest number of students – doctoral students, postgraduate students – engineers and researchers without restricting or generalizing the results.

Table of Contents:
Introduction xi Familiarization with Semi-normed Spaces xv Notations xvii Chapter 1 Prerequisites 1 1.1 Sets, mappings, orders 1 1.2 Countability 3 1.3 Construction of R 4 1.4 Properties of R 5 Part 1 Semi-normed Spaces 9 Chapter 2 Semi-normed Spaces 11 2.1 Definition of semi-normed spaces 11 2.2 Convergent sequences 15 2.3 Bounded, open and closed sets 17 2.4 Interior, closure, balls and semi-balls 21 2.5 Density, separability 23 2.6 Compact sets 25 2.7 Connected and convex sets 30 Chapter 3 Comparison of Semi-normed Spaces 33 3.1 Equivalent families of semi-norms 33 3.2 Topological equalities and inclusions 34 3.3 Topological subspaces 39 3.4 Filtering families of semi-norms 43 3.5 Sums of sets 46 Chapter 4 Banach, Fréchet and Neumann Spaces 49 4.1 Metrizable spaces 49 4.2 Properties of sets in metrizable spaces 51 4.3 Banach, Fréchet and Neumann spaces 55 4.4 Compacts sets in Fréchet spaces 57 4.5 Properties of R 58 4.6 Convergent sequences 60 4.7 Sequential completion of a semi-normed space 62 Chapter 5 Hilbert Spaces 65 5.1 Hilbert spaces 65 5.2 Projection in a Hilbert space 68 5.3 The space Rd 70 Chapter 6 Product, Intersection, Sum and Quotient of Spaces 73 6.1 Product of semi-normed spaces 73 6.2 Product of a semi-normed space by itself 78 6.3 Intersection of semi-normed spaces 80 6.4 Sum of semi-normed spaces 83 6.5 Direct sum of semi-normed spaces 89 6.6 Quotient space 93 Part 2 Continuous Mappings 95 Chapter 7 Continuous Mappings 97 7.1 Continuous mappings 97 7.2 Continuity and change of topology or restriction 100 7.3 Continuity of composite mappings 102 7.4 Continuous semi-norms 102 7.5 Continuous linear mappings 104 7.6 Continuous multilinear mappings 107 7.7 Some continuous mappings 111 Chapter 8 Images of Sets Under Continuous Mappings 115 8.1 Images of open and closed sets 115 8.2 Images of dense, separable and connected sets 117 8.3 Images of compact sets 119 8.4 Images under continuous linear mappings 121 8.5 Continuous mappings in compact sets 123 8.6 Continuous real mappings 124 8.7 Compacting mappings 125 Chapter 9 Properties of Mappings in Metrizable Spaces 129 9.1 Continuous mappings in metrizable spaces 129 9.2 Banach’s fixed point theorem 133 9.3 Baire’s theorem 134 9.4 Open mapping theorem 136 9.5 Banach–Schauder’s continuity theorem 138 9.6 Closed graph theorem 139 Chapter 10 Extension of Mappings, Equicontinuity 141 10.1 Extension of equalities by continuity 141 10.2 Continuous extension of mappings 142 10.3 Equicontinuous families of mappings 146 10.4 Banach–Steinhaus equicontinuity theorem 148 Chapter 11 Compactness in Mapping Spaces 153 11.1 The spaces F(X; F) and C(X; F)-pt 153 11.2 Zorn’s lemma 154 11.3 Compactness in F(X; F) 157 11.4 An Ascoli compactness theorem in C(X; F)-pt 161 Chapter 12 Spaces of Linear or Multilinear Mappings 163 12.1 The space L(E; F) 163 12.2 Bounded sets in L(E; F) 165 12.3 Sequential completeness of L(E; F) when E is metrizable 167 12.4 Semi-norms and norm on L(E; F) when E isnormed 169 12.5 Continuity of the composition of linear mappings 171 12.6 Inversibility in the neighborhood of an isomorphism 174 12.7 The space Ld(E1 × ··· × Ed; F) 178 12.8 Separation of the variables of a multilinear mapping 181 Part 3 Weak Topologies 187 Chapter 13 Duality 189 13.1 Dual 189 13.2 Dual of a metrizable or normed space 193 13.3 Dual of a Hilbert space 196 13.4 Extraction of ∗ weakly converging subsequences 199 13.5 Continuity of the bilinear form of duality 203 13.6 Dual of a product 205 13.7 Dual of a direct sum 206 Chapter 14 Dual of a Subspace 209 14.1 Hahn–Banach theorem 209 14.2 Corollaries of the Hahn–Banach theorem 211 14.3 Characterization of a dense subspace 212 14.4 Dual of a subspace 213 14.5 Dual of an intersection 215 14.6 Dangerous identifications 216 Chapter 15 Weak Topology 221 15.1 Weak topology 221 15.2 Weak continuity and topological inclusions 224 15.3 Weak topology of a product 225 15.4 Weak topology of an intersection 226 15.5 Norm and semi-norms of a weak limit 228 Chapter 16 Properties of Sets for the Weak Topology 231 16.1 Banach–Mackey theorem (weakly bounded sets) 231 16.2 Gauge of a convex open set 233 16.3 Mazur’s theorem (weakly closed convex sets) 235 16.4 ¢Smulian’s theorem (weakly compact sets) 237 16.5 Semi-weak continuity of a bilinear mapping 240 Chapter 17 Reflexivity 243 17.1 Reflexive spaces 243 17.2 Sequential completion of a semi-reflexive space 247 17.3 Prereflexivity of metrizable spaces 248 17.4 Reflexivity of Hilbert spaces 250 17.5 Reflexivity of uniformly convex Banach spaces 252 17.6 A property of the combinations of linear forms 256 17.7 Characterizations of semi-reflexivity 257 17.8 Reflexivity of a subspace 261 17.9 Reflexivity of the image of a space 261 17.10 Reflexivity of the dual 263 Chapter 18 Extractable Spaces 265 18.1 Extractable spaces 265 18.2 Extractability of Hilbert spaces 266 18.3 Extractability of semi-reflexive spaces 267 18.4 Extractability of a subspace or of the image of a space 269 18.5 Extractability of a product or of a sum of spaces 270 18.6 Extractability of an intersection of spaces 271 18.7 Sequential completion of extractable spaces 271 Part 4 Differential Calculus 273 Chapter 19 Differentiable Mappings 275 19.1 Differentiable mappings 275 19.2 Differentiality, continuity and linearity 277 19.3 Differentiation and change of topology or restriction 279 19.4 Mean value theorem 281 19.5 Bounds on a real differentiable mapping 284 19.6 Differentiation of a composite mapping 286 19.7 Differential of an inverse mapping 289 19.8 Inverse mapping theorem 290 Chapter 20 Differentiation of Multivariable Mappings 295 20.1 Partial differentiation 295 20.2 Differentiation of a multilinear or multi-component mapping 298 20.3 Differentiation of a composite multilinear mapping 300 Chapter 21 Successive Differentiations 303 21.1 Successive differentiations 303 21.2 Schwarz’s symmetry principle 305 21.3 Successive differentiations of a composite mapping 308 Chapter 22 Derivation of Functions of One Real Variable 313 22.1 Derivative of a function of one real variable 313 22.2 Derivative of a real function of one real variable 315 22.3 Leibniz formula 319 22.4 Derivatives of the power, logarithm and exponential functions 320 Bibliography 325 Cited Authors 331 Index 335

About the Author :
Jacques Simon, CNRS, France.