History of Functional Analysis (eBook)

Front Cover 1
History of Functional Analysis 4
Copyright Page 5
TABLE OF CONTENTS 6
INTRODUCTION 8
CHAPTER I: LINEAR DIFFERENTIAL EQUATIONS AND THE STURM-LIOUVILLE PROBLEM 16
§1. Differential equations and partial differential equations in the XVIII th century 16
§2. Fourier expansions 18
§3. The Sturm - Liouville theory 23
CHAPTER II: THE "CRYPTO-INTEGRAL" EQUATIONS 29
§1. The method of successive approximations 29
§2. Partial differential equations in the XIX th century 33
§3. The beginnings of potential theory 37
§4. The Dirichlet Principle 42
§5. The Beer - Neumann method 46
CHAPTER III: THE EQUATION OF VIBRATING MEMBRANES 54
§1. H.A. Schwarz's 1885 paper 54
§2. The contributions of Poincaré 63
CHAPTER IV: THE IDEA OF INFINITE DIMENSION 78
§1. Linear algebra in the XIX th century 78
§2. Infinite determinants 82
§3. Groping towards function spaces 86
§4 . The passage " from finiteness to infinity" 94
CHAPTER V: THE CRUCIAL YEARS AND THE DEFINITION OF HILBERT SPACE 104
§1. Fredholm's discovery 104
§2. The contributions of Hilbert 112
§3. The confluence of Geometry, Topology and Analysis 122
CHAPTER VI: DUALITY AND THE DEFINITION OF NORMED SPACES 128
§1. The search for continuous linear functionals 128
§2. The Lp and lp, spaces 131
§3. The birth of normed spaces and the Hahn - Banach theorem 135
§4. The method of the gliding hump and Baire category 145
CHAPTER VII: SPECTRAL THEORY AFTER 1900 151
§1. F. Riesz's theory of compact operators 151
§2. The spectral theory of Hilbert 155
§3. The work of Weyl and Carleman 167
§4. The spectral theory of von Neumann 178
§5. Bsnach slgebras 189
§6. Later developments 197
CHAPTER VIII: LOCALLY CONVEX SPACES AND THE THEORY OF DISTRIBUTIONS 217
§1. Weak convergence and weak topology 217
§2. Locally convex spaces . 222
§3 . The theory of distributions 228
CHAPTER IX: APPLICATIONS OF FUNCTIONAL ANALYSIS TO DIFFERENTIAL AND PARTIAL DIFFERENTIAL EQUATIONS 240
§1. Fixed point theorems 240
§2. Carleman operators and generalized eigenvectors 245
§3. Boundary problems for ordinary diffenential equations 250
§4. Sobolev spaces and a priori inequalities 255
§5. Elementary soultions, paramentrices and pseudo-differential operators 259
REFERENCES 287
AUTHOR INDEX 306
SUBJECT INDEX 313