Mathematical analysis during the 20th century / Jean-Paul Pier.

Format:

Book

Author/Creator:

Pier, Jean-Paul, 1933-

Language:

English

Subjects (All):

Mathematical analysis--History--20th century.

Mathematical analysis.

History.

Physical Description:

x, 428 pages ; 25 cm

Other Title:

Mathematical analysis during the twentieth century

Place of Publication:

Oxford ; New York : Oxford University Press, 2001.

Summary:

Published under the Clarendon Press imprint, this book covers the 20th Century evolution of essential ideas in mathematical analysis, a field that since the times of Newton and Leibnitz has been one of the most important and presitigious in mathematics. Each chapter features a comprehensive first part on developments during the period 1900-1950, then provides outlooks on achievements during the last part of the century. Chapters include original quotations from outstanding mathematicians and the book includes a bibliography of almost 1000 entries.

Contents:

1.1.1 The great classics on analysis 2

1.1.2 The changing object of analysis 4

1.2 Main streams in a turbulent activity 12

1.2.1 The question of subdividing mathematical analysis 12

1.2.2 How to organize the subject 16

2 General Topology 27

2.1 Evolution 1900-1950 27

2.1.1 Topological axiomatizations 27

2.1.2 Topological algebra 37

2.1.3 Filtrations 40

2.1.4 Dimension theory 42

2.1.5 Complementary inputs 42

2.2 Flashes 1950-2000 44

2.2.1 An accomplished subject 44

2.2.2 Generalized topological concepts 45

3 Integration and Measure 47

3.1 Evolution 1900-1950 47

3.1.1 Lebesgue integration 47

3.1.2 The general concept of measure 55

3.1.3 Paradoxical decomposition 61

3.1.4 Period of consolidation 61

3.2 Flashes 1950-2000 63

3.2.1 Standing problems 63

3.2.2 Abstract formulations 64

3.2.3 Generalized Riemann integrals 68

3.2.4 Outlook 69

4 Functional analysis 73

4.1 Evolution 1900-1950 73

4.1.1 New objectives 73

4.1.2 Theory of integral equations 79

4.1.3 Banach spaces 86

4.1.4 Hilbert spaces 92

4.1.5 von Neumann algebras 95

4.1.6 Banach algebras 98

4.1.7 Distributions 101

4.2 Flashes 1950-2000 106

4.2.1 Topological vector spaces 106

4.2.2 Extension of Weierstra[beta]'s theorem 108

4.2.3 Frechet spaces, Schwartz spaces, Sobolev spaces 108

4.2.4 Banach space properties 110

4.2.5 Hilbert space properties 113

4.2.6 Banach algebra and C*-algebra properties 114

4.2.7 Approximation properties 117

4.2.8 Nuclearity 117

4.2.9 von Neumann algebra properties 123

4.2.10 Specific topics 126

5 Harmonic analysis 129

5.1 Evolution 1900-1950 129

5.1.1 Fourier series 129

5.1.2 Invariant measures 137

5.1.3 Almost periodic functions 140

5.1.4 Uniqueness of invariant measures 141

5.1.5 Convolutions 142

5.1.6 An evolution linked to the history of physics 145

5.1.7 Representation theory 150

5.1.8 Structural properties of topological groups 154

5.1.9 Positive-definite functions 155

5.1.10 Harmonic synthesis 157

5.1.11 Metric locally compact Abelian groups 160

5.2 Flashes 1950-2000 163

5.2.1 Fourier transforms 163

5.2.2 Convolution properties 166

5.2.3 Group representations 166

5.2.4 Remarkable Banach algebras of functions on a locally compact group 169

5.2.5 Specific sets 170

5.2.6 Specific groups 171

5.2.7 Harmonic analysis on semigroups 173

5.2.8 Wavelets 174

5.2.9 Generalized actions 176

6 Lie groups 179

6.1 Evolution 1900-1950 179

6.1.1 Lie groups and Lie algebras 179

6.1.2 Symmetric Riemannian spaces 185

6.1.3 Hilbert's problem for Lie groups 187

6.1.4 Representations of Lie groups 188

6.2 Flashes 1950-2000 189

6.2.1 The wide range of Lie group theory 189

6.2.2 Solution of Hilbert's problem on Lie groups 191

6.2.3 Ergodicity problems 191

6.2.4 Specific classes of Lie groups 192

6.2.5 Extensions of Lie group theory 196

7 Theory of functions and analytic geometry 199

7.1 Evolution 1900-1950 200

7.1.1 The nineteenth century continued 200

7.1.2 Potential theory 209

7.1.3 Conformal mappings 211

7.1.4 Towards a theory of several complex variables 212

7.2 Flashes 1950-2000 215

7.2.1 Accomplishments on previous topics 215

7.2.2 Hardy spaces 219

7.2.3 The dominance of the theory of several complex variables 223

7.2.4 Iteration problems 227

8 Ordinary and Partial Differential Equations 229

8.1 Evolution 1900-1950 229

8.1.1 New trends for classical problems 229

8.1.2 Fixed point properties 231

8.1.3 From the ordinary differential case to the partial differential case 232

8.2 Flashes 1950-2000 236

8.2.1 Differential equations 236

8.2.2 Partial differential equations 238

8.2.3 Tentacular subjects 247

9 Algebraic topology 255

9.1 Evolution 1900-1950 255

9.1.1 The origins of algebraic topology 255

9.1.2 Simplicial theories 261

9.1.3 Homotopy theory 267

9.1.4 Fibres and fibrations 270

9.1.5 The breakthroughs due to Eilenberg, MacLane, and Leray 271

9.2 Flashes 1950-2000 275

9.2.1 The power of the machinery 275

9.2.2 Generalizations 278

10 Differential topology 280

10.1 Evolution 1900-1950 280

10.1.1 The beginning of the century 280

10.1.2 E. Cartan's work 283

10.1.3 Tensor products and exterior differentials 284

10.1.4 Morse theory 287

10.1.5 Whitney's work 288

10.1.6 De Rham's work 291

10.1.7 Hodge theory 293

10.1.8 The framing of the subject 297

10.2 Flashes 1950-2000 299

10.2.1 The status of differentiable manifolds 299

10.2.2 Foliations 302

10.2.3 New objectives 303

10.2.4 From Poincare's heritage 307

10.2.5 Global analysis 310

11 Probability 314

11.1 Evolution 1900-1950 314

11.1.1 First results 314

11.1.2 Brownian motion 318

11.1.3 Ergodicity 320

11.1.4 Probabilities as measures 321

11.1.5 Stochastic integrals 324

11.2 Flashes 1950-2000 325

11.2.1 Probability theory, a part of analysis 325

11.2.2 Dynamical systems and ergodicity 326

11.2.3 Entropy 328

11.2.4 Stochastic processes 328

12 Algebraic geometry 331

12.1 Evolution 1900-1950 331

12.1.1 Algebraic geometry and number theory 331

12.1.2 The Mordell conjecture 334

12.1.3 Transcendence and prime numbers 336

12.1.4 The Riemann conjecture 338

12.2 Flashes 1950-2000 339

12.2.1 Arithmetical properties 339

12.2.2 Investigations on transcendental numbers 340

12.2.3 A central object of study 341

12.2.4 Etale cohomology 345

12.2.5 The general Riemann-Roch theorems 345

12.2.6 K-theory 346

12.2.7 Further studies 349.

Notes:

Includes bibliographical references (pages [350]-409) and indexes.

ISBN:

0198503946

OCLC:

46619295