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Theory of functions of a real variable.

Theory of functions of a real variable. Shlomo Sternberg May 10, 2005. 2. Introduction. I have taught the beginning graduate course in real variables and functional analysis three times in the last five years, and this book is the result. The course assumes that the student has seen the basics of real variable Theory and point set topology. The elements of the topology of metrics spaces are presented (in the nature of a rapid review) in Chapter I. The course itself consists of two parts: 1) measure Theory and integration, and 2) Hilbert space Theory , especially the spectral theorem and its applications. In Chapter II I do the basics of Hilbert space Theory , what I can do without measure Theory or the Lebesgue integral. The hero here (and perhaps for the first half of the course) is the Riesz representation theorem . Included is the spectral theorem for compact self-adjoint operators and applications of this theorem to elliptic partial differential equations.

3 the spectral theorem to quantum mechanics and quantum chemistry. Chapter XIII is a brief introduction to the Lax-Phillips theory of scattering.

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Transcription of Theory of functions of a real variable.

1 Theory of functions of a real variable. Shlomo Sternberg May 10, 2005. 2. Introduction. I have taught the beginning graduate course in real variables and functional analysis three times in the last five years, and this book is the result. The course assumes that the student has seen the basics of real variable Theory and point set topology. The elements of the topology of metrics spaces are presented (in the nature of a rapid review) in Chapter I. The course itself consists of two parts: 1) measure Theory and integration, and 2) Hilbert space Theory , especially the spectral theorem and its applications. In Chapter II I do the basics of Hilbert space Theory , what I can do without measure Theory or the Lebesgue integral. The hero here (and perhaps for the first half of the course) is the Riesz representation theorem . Included is the spectral theorem for compact self-adjoint operators and applications of this theorem to elliptic partial differential equations.

2 The pde material follows closely the treatment by Bers and Schecter in Partial Differential Equations by Bers, John and Schecter AMS (1964). Chapter III is a rapid presentation of the basics about the Fourier transform. Chapter IV is concerned with measure Theory . The first part follows Caratheodory's classical presentation. The second part dealing with Hausdorff measure and di- mension, Hutchinson's theorem and fractals is taken in large part from the book by Edgar, Measure Theory , Topology, and Fractal Geometry Springer (1991). This book contains many more details and beautiful examples and pictures. Chapter V is a standard treatment of the Lebesgue integral. Chapters VI, and VIII deal with abstract measure Theory and integration. These chapters basically follow the treatment by Loomis in his Abstract Har- monic Analysis. Chapter VII develops the Theory of Wiener measure and Brownian motion following a classical paper by Ed Nelson published in the Journal of Mathemat- ical Physics in 1964.

3 Then we study the idea of a generalized random process as introduced by Gelfand and Vilenkin, but from a point of view taught to us by Dan Stroock. The rest of the book is devoted to the spectral theorem . We present three proofs of this theorem . The first, which is currently the most popular, derives the theorem from the Gelfand representation theorem for Banach algebras. This is presented in Chapter IX (for bounded operators). In this chapter we again follow Loomis rather closely. In Chapter X we extend the proof to unbounded operators, following Loomis and Reed and Simon Methods of Modern Mathematical Physics. Then we give Lorch's proof of the spectral theorem from his book Spectral Theory . This has the flavor of complex analysis. The third proof due to Davies, presented at the end of Chapter XII replaces complex analysis by almost complex analysis. The remaining chapters can be considered as giving more specialized in- formation about the spectral theorem and its applications.

4 Chapter XI is de- voted to one parameter semi-groups, and especially to Stone's theorem about the infinitesimal generator of one parameter groups of unitary transformations. Chapter XII discusses some theorems which are of importance in applications of 3. the spectral theorem to quantum mechanics and quantum chemistry. Chapter XIII is a brief introduction to the Lax-Phillips Theory of scattering. 4. Contents 1 The topology of metric spaces 13. Metric spaces .. 13. Completeness and completion.. 16. Normed vector spaces and Banach spaces.. 17. Compactness.. 18. Total Boundedness.. 18. Separability.. 19. Second Countability.. 20. Conclusion of the proof of theorem .. 20. Dini's lemma.. 21. The Lebesgue outer measure of an interval is its length.. 21. Zorn's lemma and the axiom of choice.. 23. The Baire category theorem .. 24. Tychonoff's theorem .. 24. Urysohn's lemma.. 25. The Stone-Weierstrass theorem .. 27. Machado's theorem .. 30.

5 The Hahn-Banach theorem .. 32. The Uniform Boundedness Principle.. 35. 2 Hilbert Spaces and Compact operators. 37. Hilbert space.. 37. Scalar products.. 37. The Cauchy-Schwartz inequality.. 38. The triangle inequality .. 39. Hilbert and pre-Hilbert spaces.. 40. The Pythagorean theorem .. 41. The theorem of Apollonius.. 42. The theorem of Jordan and von Neumann.. 42. Orthogonal projection.. 45. The Riesz representation theorem .. 47. What is L2 (T)? .. 48. Projection onto a direct sum.. 49. Projection onto a finite dimensional subspace.. 49. 5. 6 CONTENTS. Bessel's inequality.. 49. Parseval's equation.. 50. Orthonormal bases.. 50. Self-adjoint transformations.. 51. Non-negative self-adjoint transformations.. 52. Compact self-adjoint transformations.. 54. Fourier's Fourier series.. 57. Proof by integration by parts.. 57. d Relation to the operator dx .. 60. G arding's inequality, special case.. 62. The Heisenberg uncertainty principle.

6 64. The Sobolev Spaces.. 67. G arding's inequality.. 72. Consequences of G arding's inequality.. 76. Extension of the basic lemmas to manifolds.. 79. Example: Hodge Theory .. 80. The resolvent.. 83. 3 The Fourier Transform. 85. Conventions, especially about 2 .. 85. Convolution goes to multiplication.. 86. Scaling.. 86. Fourier transform of a Gaussian is a Gaussian.. 86. The multiplication formula.. 88. The inversion formula.. 88. Plancherel's theorem .. 88. The Poisson summation formula.. 89. The Shannon sampling theorem .. 90. The Heisenberg Uncertainty Principle.. 91. Tempered distributions.. 92. Examples of Fourier transforms of elements of S 0 .. 93. 4 Measure Theory . 95. Lebesgue outer measure.. 95. Lebesgue inner measure.. 98. Lebesgue's definition of measurability.. 98. Caratheodory's definition of measurability.. 102. Countable additivity.. 104. -fields, measures, and outer measures.. 108. Constructing outer measures, Method I.

7 109. A pathological example.. 110. Metric outer measures.. 111. Constructing outer measures, Method II.. 113. An example.. 114. Hausdorff measure.. 116. Hausdorff dimension.. 117. CONTENTS 7. Push forward.. 119. The Hausdorff dimension of fractals .. 119. Similarity dimension.. 119. The string model.. 122. The Hausdorff metric and Hutchinson's theorem .. 124. Affine examples .. 126. The classical Cantor set.. 126. The Sierpinski Gasket .. 128. Moran's theorem .. 129. 5 The Lebesgue integral. 133. Real valued measurable functions .. 134. The integral of a non-negative function.. 134. Fatou's lemma.. 138. The monotone convergence theorem .. 140. The space L1 (X, R).. 140. The dominated convergence theorem .. 143. Riemann integrability.. 144. The Beppo - Levi theorem .. 145. L1 is complete.. 146. Dense subsets of L1 (R, R).. 147. The Riemann-Lebesgue Lemma.. 148. The Cantor-Lebesgue theorem .. 150. Fubini's theorem .. 151. Product -fields.

8 151. -systems and -systems.. 152. The monotone class theorem .. 153. Fubini for finite measures and bounded functions .. 154. Extensions to unbounded functions and to -finite 6 The Daniell integral. 157. The Daniell Integral .. 157. Monotone class theorems.. 160. Measure.. 161. H older, Minkowski , Lp and Lq .. 163. k k is the essential sup norm.. 166. The Radon-Nikodym theorem .. 167. The dual space of Lp .. 170. The variations of a bounded functional.. 171. Duality of Lp and Lq when (S) < .. 172. The case where (S) = .. 173. Integration on locally compact Hausdorff spaces.. 175. Riesz representation theorems.. 175. Fubini's theorem .. 176. The Riesz representation theorem redux.. 177. Statement of the theorem .. 177. 8 CONTENTS. Propositions in topology.. 178. Proof of the uniqueness of the restricted to B(X).. 180. Existence.. 180. Definition.. 180. Measurability of the Borel sets.. 182. Compact sets have finite measure.. 183.

9 Interior regularity.. 183. Conclusion of the proof.. 184. 7 Wiener measure, Brownian motion and white noise. 187. Wiener measure.. 187. The Big Path Space.. 187. The heat equation.. 189. Paths are continuous with probability one.. 190. Embedding in S 0 .. 194. Stochastic processes and generalized stochastic processes.. 195. Gaussian measures.. 196. Generalities about expectation and variance.. 196. Gaussian measures and their variances.. 198. The variance of a Gaussian with density.. 199. The variance of Brownian motion.. 200. The derivative of Brownian motion is white noise.. 202. 8 Haar measure. 205. Examples.. 206. Rn .. 206. Discrete groups.. 206. Lie groups.. 206. Topological facts.. 211. Construction of the Haar integral.. 212. Uniqueness.. 216. (G) < if and only if G is compact.. 218. The group algebra.. 218. The involution.. 220. The modular function.. 220. Definition of the involution.. 222. Relation to convolution.

10 223. Banach algebras with involutions.. 223. The algebra of finite measures.. 223. Algebras and coalgebras.. 224. Invariant and relatively invariant measures on homogeneous CONTENTS 9. 9 Banach algebras and the spectral theorem . 231. Maximal ideals.. 232. Existence.. 232. The maximal spectrum of a ring.. 232. Maximal ideals in a commutative algebra.. 233. Maximal ideals in the ring of continuous functions .. 234. Normed algebras.. 235. The Gelfand representation.. 236. Invertible elements in a Banach algebra form an open set. 238. The Gelfand representation for commutative Banach al- gebras.. 241. The spectral radius.. 241. The generalized Wiener theorem .. 242. Self-adjoint algebras.. 244. An important generalization.. 247. An important application.. 248. The Spectral theorem for Bounded Normal Operators, Func- tional Calculus Form.. 249. Statement of the theorem .. 250. SpecB (T ) = SpecA (T ).. 251. A direct proof of the spectral theorem .


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