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