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, i.e. 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. 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 b ook 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 Lo om is 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. Then we study the idea of a generalized random proc ess 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 pro of 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. 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 topol ogy of metric spaces 13 1.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Completeness and completion. . . . . . . . . . . . . . . . . . . . . 16 1.3 Normed vector spaces and Banach spaces. . . . . . . . . . . . . . 17 1.4 Compactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Total Boundedness. . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Separability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.7 Second Countability. . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.8 Conclusion of the proof of Theorem 1.5.1. . . . . . . . . . . . . . 20 1.9 Dini’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.10 The Lebesgue outer measure of an interval is its length. . . . . . 21 1.11 Zorn’s lemma and the axiom of choice. . . . . . . . . . . . . . . . 23 1.12 The Baire category theorem. . . . . . . . . . . . . . . . . . . . . 24 1.13 Tychonoff’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.14 Urysohn’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.15 The Stone-Weierstrass theorem. . . . . . . . . . . . . . . . . . . . 27 1.16 Machado’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.17 The Hahn-Banach theorem. . . . . . . . . . . . . . . . . . . . . . 32 1.18 The Uniform Boundedness Principle. . . . . . . . . . . . . . . . . 35 2 Hilbert Spaces and Compact operators. 37 2.1 Hilbert space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.1 Scalar products. . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.2 The Cauchy-Schwartz inequality. . . . . . . . . . . . . . . 38 2.1.3 The triangle inequality . . . . . . . . . . . . . . . . . . . . 39 2.1.4 Hilbert and pre-Hilbert spaces. . . . . . . . . . . . . . . . 40 2.1.5 The Pythagorean theorem. . . . . . . . . . . . . . . . . . 41 2.1.6 The theorem of Apollonius. . . . . . . . . . . . . . . . . . 42 2.1.7 The theorem of Jordan and von Neumann. . . . . . . . . 42 2.1.8 Orthogonal projection. . . . . . . . . . . . . . . . . . . . . 45 2.1.9 The Riesz representation theorem. . . . . . . . . . . . . . 47 2.1.10 What is L 2 (T)? . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.11 Projection onto a direct sum. . . . . . . . . . . . . . . . . 49 2.1.12 Projection onto a finite dimensional subspace. . . . . . . . 49 5 6 CONTENTS 2.1.13 Bessel’s inequality. . . . . . . . . . . . . . . . . . . . . . . 49 2.1.14 Parseval’s equation. . . . . . . . . . . . . . . . . . . . . . 50 2.1.15 Orthonormal bases. . . . . . . . . . . . . . . . . . . . . . 50 2.2 Self-adjoint transformations. . . . . . . . . . . . . . . . . . . . . . 51 2.2.1 Non-negative self-adjoint transformations. . . . . . . . . . 52 2.3 Compact self-adjoint transformations. . . . . . . . . . . . . . . . 54 2.4 Fourier’s Fourier series. . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.1 Proof by integration by parts. . . . . . . . . . . . . . . . . 57 2.4.2 Relation to the operator d dx . . . . . . . . . . . . . . . . . . 60 2.4.3 G˚arding’s inequality, special case. . . . . . . . . . . . . . . 62 2.5 The Heisenberg uncertainty principle. . . . . . . . . . . . . . . . 64 2.6 The Sobolev Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.7 G˚arding’s inequality. . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.8 Consequences of G˚arding’s inequality. . . . . . . . . . . . . . . . 76 2.9 Extension of the basic lemmas to m anifolds. . . . . . . . . . . . . 79 2.10 Example: Hodge theory. . . . . . . . . . . . . . . . . . . . . . . . 80 2.11 The resolvent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3 The Fourier Transform. 85 3.1 Conventions, especially about 2π. . . . . . . . . . . . . . . . . . . 85 3.2 Convolution goes to multiplication. . . . . . . . . . . . . . . . . . 86 3.3 Scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4 Fourier transform of a Gaussian is a Gaussian. . . . . . . . . . . 86 3.5 The multiplication formula. . . . . . . . . . . . . . . . . . . . . . 88 3.6 The inversion formula. . . . . . . . . . . . . . . . . . . . . . . . . 88 3.7 Plancherel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.8 The Poisson summation formula. . . . . . . . . . . . . . . . . . . 89 3.9 The Shannon sampling theorem. . . . . . . . . . . . . . . . . . . 90 3.10 The Heisenberg Uncertainty Principle. . . . . . . . . . . . . . . . 91 3.11 Tempered distributions. . . . . . . . . . . . . . . . . . . . . . . . 92 3.11.1 Examples of Fourier transforms of elements of S  . . . . . . 93 4 Measure theory. 95 4.1 Lebesgue outer measure. . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Lebesgue inner measure. . . . . . . . . . . . . . . . . . . . . . . . 98 4.3 Lebesgue’s definition of measurability. . . . . . . . . . . . . . . . 98 4.4 Caratheodory’s definition of measurability. . . . . . . . . . . . . . 102 4.5 Countable additivity. . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.6 σ-fields, measures, and outer measures. . . . . . . . . . . . . . . . 108 4.7 Constructing outer measures, Method I. . . . . . . . . . . . . . . 109 4.7.1 A pathological example. . . . . . . . . . . . . . . . . . . . 110 4.7.2 Metric outer measures. . . . . . . . . . . . . . . . . . . . . 111 4.8 Constructing outer measures, Method II. . . . . . . . . . . . . . . 113 4.8.1 An example. . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.9 Hausdorff measure. . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.10 Hausdorff dimension. . . . . . . . . . . . . . . . . . . . . . . . . . 117 CONTENTS 7 4.11 Push forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.12 The Hausdorff dimension of fractals . . . . . . . . . . . . . . . . 119 4.12.1 Similarity dimension. . . . . . . . . . . . . . . . . . . . . . 119 4.12.2 The string model. . . . . . . . . . . . . . . . . . . . . . . 122 4.13 The Hausdorff metric and Hutchinson’s theorem. . . . . . . . . . 124 4.14 Affine examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.14.1 The classical Cantor set. . . . . . . . . . . . . . . . . . . . 126 4.14.2 The Sierpinski Gasket . . . . . . . . . . . . . . . . . . . . 128 4.14.3 Moran’s theorem . . . . . . . . . . . . . . . . . . . . . . . 129 5 The Lebesgue integral. 133 5.1 Real valued measurable functions. . . . . . . . . . . . . . . . . . 134 5.2 The integral of a non-negative function. . . . . . . . . . . . . . . 134 5.3 Fatou’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.4 The monotone convergence theorem. . . . . . . . . . . . . . . . . 140 5.5 The space L 1 (X, R). . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.6 The dominated convergence theorem. . . . . . . . . . . . . . . . . 143 5.7 Riemann integrability. . . . . . . . . . . . . . . . . . . . . . . . . 144 5.8 The Beppo - Levi theorem. . . . . . . . . . . . . . . . . . . . . . 145 5.9 L 1 is complete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.10 Dense subsets of L 1 (R, R). . . . . . . . . . . . . . . . . . . . . . 147 5.11 The Riemann-Lebesgue Lemma. . . . . . . . . . . . . . . . . . . 148 5.11.1 The Cantor-Lebes gue theorem. . . . . . . . . . . . . . . . 150 5.12 Fubini’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.12.1 Product σ-fields. . . . . . . . . . . . . . . . . . . . . . . . 151 5.12.2 π-system s and λ-systems. . . . . . . . . . . . . . . . . . . 152 5.12.3 The monotone class theorem. . . . . . . . . . . . . . . . . 153 5.12.4 Fubini for finite measures and bounded functions. . . . . 154 5.12.5 Extensions to unbounded functions and to σ-finite measures.156 6 The Daniell integral. 157 6.1 The Daniell Integral . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Monotone class theorems. . . . . . . . . . . . . . . . . . . . . . . 160 6.3 Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4 H¨older, Minkowski , L p and L q . . . . . . . . . . . . . . . . . . . . 163 6.5  ·  ∞ is the essential sup norm. . . . . . . . . . . . . . . . . . . . 166 6.6 The Radon-Nikodym Theorem. . . . . . . . . . . . . . . . . . . . 167 6.7 The dual space of L p . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.7.1 The variations of a bounded functional. . . . . . . . . . . 171 6.7.2 Duality of L p and L q when µ(S) < ∞. . . . . . . . . . . . 172 6.7.3 The case where µ(S) = ∞. . . . . . . . . . . . . . . . . . 173 6.8 Integration on locally compact Hausdorff spaces. . . . . . . . . . 175 6.8.1 Riesz representation theorems. . . . . . . . . . . . . . . . 175 6.8.2 Fubini’s theorem. . . . . . . . . . . . . . . . . . . . . . . . 176 6.9 The Riesz representation theorem redux. . . . . . . . . . . . . . . 177 6.9.1 Statement of the theorem. . . . . . . . . . . . . . . . . . . 177 8 CONTENTS 6.9.2 Prop os itions in topology. . . . . . . . . . . . . . . . . . . 178 6.9.3 Proof of the uniqueness of the µ restricted to B(X). . . . 180 6.10 Existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.10.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.10.2 Measurability of the Borel sets. . . . . . . . . . . . . . . . 182 6.10.3 Compact sets have finite measure. . . . . . . . . . . . . . 183 6.10.4 Interior regularity. . . . . . . . . . . . . . . . . . . . . . . 183 6.10.5 Conclusion of the pro of. . . . . . . . . . . . . . . . . . . . 184 7 Wiener measure, Brownian motion and white noise. 187 7.1 Wiener measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.1.1 The Big Path Space. . . . . . . . . . . . . . . . . . . . . . 187 7.1.2 The heat equation. . . . . . . . . . . . . . . . . . . . . . . 189 7.1.3 Paths are continuous with probability one. . . . . . . . . 190 7.1.4 Embedding in S  . . . . . . . . . . . . . . . . . . . . . . . . 194 7.2 Stochastic processes and generalized stochastic processes. . . . . 195 7.3 Gaussian measures. . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.3.1 Generalities about expectation and variance. . . . . . . . 196 7.3.2 Gaussian measures and their variances. . . . . . . . . . . 198 7.3.3 The variance of a Gaussian with density. . . . . . . . . . . 199 7.3.4 The variance of Brownian motion. . . . . . . . . . . . . . 200 7.4 The derivative of Brownian motion is white noise. . . . . . . . . . 202 8 Haar measure. 205 8.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.1.1 R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.1.2 Discrete groups. . . . . . . . . . . . . . . . . . . . . . . . 206 8.1.3 Lie groups. . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.2 Topological facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.3 Construction of the Haar integral. . . . . . . . . . . . . . . . . . 212 8.4 Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.5 µ(G) < ∞ if and only if G is compact. . . . . . . . . . . . . . . . 218 8.6 The group algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.7 The involution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.7.1 The modular function. . . . . . . . . . . . . . . . . . . . . 220 8.7.2 Definition of the involution. . . . . . . . . . . . . . . . . . 222 8.7.3 Relation to convolution. . . . . . . . . . . . . . . . . . . . 223 8.7.4 Banach algebras with involutions. . . . . . . . . . . . . . 223 8.8 The algebra of finite measures. . . . . . . . . . . . . . . . . . . . 223 8.8.1 Algebras and coalgebras. . . . . . . . . . . . . . . . . . . . 224 8.9 Invariant and relatively invariant measures on homogeneous spaces.225 CONTENTS 9 9 Banach algebras and the spectral theorem. 231 9.1 Maximal ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 9.1.1 Existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 9.1.2 The maximal spectrum of a ring. . . . . . . . . . . . . . . 232 9.1.3 Maximal ideals in a commutative algebra. . . . . . . . . . 233 9.1.4 Maximal ideals in the ring of continuous functions. . . . . 234 9.2 Normed algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.3 The Gelfand representation. . . . . . . . . . . . . . . . . . . . . . 236 9.3.1 Invertible elements in a Banach algebra form an open set. 238 9.3.2 The Gelfand representation for commutative Banach al- gebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.3.3 The spectral radius. . . . . . . . . . . . . . . . . . . . . . 241 9.3.4 The generalized Wiener theorem. . . . . . . . . . . . . . . 242 9.4 Self-adjoint algebras. . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.4.1 An important generalization. . . . . . . . . . . . . . . . . 247 9.4.2 An important application. . . . . . . . . . . . . . . . . . . 248 9.5 The Spectral Theorem for Bounded Normal Operators, Func- tional Calculus Form. . . . . . . . . . . . . . . . . . . . . . . . . 249 9.5.1 Statement of the theorem. . . . . . . . . . . . . . . . . . . 250 9.5.2 Spec B (T ) = Spec A (T ). . . . . . . . . . . . . . . . . . . . . 251 9.5.3 A direct proof of the spectral theorem. . . . . . . . . . . . 253 10 The spectral theorem. 255 10.1 Resolutions of the identity. . . . . . . . . . . . . . . . . . . . . . 256 10.2 The spectral theorem for bounded normal operators. . . . . . . . 261 10.3 Stone’s formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.4 Unbounded operators. . . . . . . . . . . . . . . . . . . . . . . . . 262 10.5 Operators and their domains. . . . . . . . . . . . . . . . . . . . . 263 10.6 The adjoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.7 Self-adjoint operators. . . . . . . . . . . . . . . . . . . . . . . . . 265 10.8 The resolvent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 10.9 The multiplication operator form of the spectral theorem. . . . . 268 10.9.1 Cyclic vectors. . . . . . . . . . . . . . . . . . . . . . . . . 269 10.9.2 The general case. . . . . . . . . . . . . . . . . . . . . . . . 271 10.9.3 The spectral theorem for unbounded self-adjoint opera- tors, multiplication operator form. . . . . . . . . . . . . . 271 10.9.4 The functional calculus. . . . . . . . . . . . . . . . . . . . 273 10.9.5 Resolutions of the identity. . . . . . . . . . . . . . . . . . 274 10.10The Riesz-Dunford calculus. . . . . . . . . . . . . . . . . . . . . . 276 10.11Lorch’s proof of the spectral theorem. . . . . . . . . . . . . . . . 279 10.11.1 Positive operators. . . . . . . . . . . . . . . . . . . . . . . 279 10.11.2 The point spectrum. . . . . . . . . . . . . . . . . . . . . . 281 10.11.3 Partition into pure types. . . . . . . . . . . . . . . . . . . 282 10.11.4 Completion of the proof. . . . . . . . . . . . . . . . . . . . 283 10.12Characterizing operators with purely continuous spectrum. . . . 287 10.13Appendix. The closed graph theorem. . . . . . . . . . . . . . . . 288 10 CONTENTS 11 Stone’s theorem 291 11.1 von Neumann’s Cayley transform. . . . . . . . . . . . . . . . . . 292 11.1.1 An elementary example. . . . . . . . . . . . . . . . . . . . 297 11.2 Equibounded semi-groups on a Frechet space. . . . . . . . . . . . 299 11.2.1 The infinitesimal generator. . . . . . . . . . . . . . . . . . 299 11.3 The differential equation . . . . . . . . . . . . . . . . . . . . . . . 301 11.3.1 The resolvent. . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.3.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 11.4 The power series expansion of the exponential. . . . . . . . . . . 309 11.5 The Hille Yosida theorem. . . . . . . . . . . . . . . . . . . . . . . 310 11.6 Contraction semigroups. . . . . . . . . . . . . . . . . . . . . . . . 313 11.6.1 Dissipation and contraction. . . . . . . . . . . . . . . . . . 314 11.6.2 A special case : exp(t(B − I)) with B ≤ 1. . . . . . . . . 316 11.7 Convergence of semigroups. . . . . . . . . . . . . . . . . . . . . . 317 11.8 The Trotter product formula. . . . . . . . . . . . . . . . . . . . . 320 11.8.1 Lie’s formula. . . . . . . . . . . . . . . . . . . . . . . . . . 320 11.8.2 Chernoff’s theorem. . . . . . . . . . . . . . . . . . . . . . 321 11.8.3 The product formula. . . . . . . . . . . . . . . . . . . . . 322 11.8.4 Commutators. . . . . . . . . . . . . . . . . . . . . . . . . 323 11.8.5 The Kato-Rellich theorem. . . . . . . . . . . . . . . . . . 323 11.8.6 Feynman path integrals. . . . . . . . . . . . . . . . . . . . 324 11.9 The Feynman-Kac formula. . . . . . . . . . . . . . . . . . . . . . 326 11.10The free Hamiltonian and the Yukawa potential. . . . . . . . . . 328 11.10.1 The Yukawa potential and the resolvent. . . . . . . . . . . 329 11.10.2 The time evolution of the free Hamiltonian. . . . . . . . . 331 12 More about the spectral theorem 333 12.1 Bound states and scattering states. . . . . . . . . . . . . . . . . . 333 12.1.1 Schwartzschild’s theorem. . . . . . . . . . . . . . . . . . . 333 12.1.2 The mean ergodic theorem . . . . . . . . . . . . . . . . . 335 12.1.3 General considerations. . . . . . . . . . . . . . . . . . . . 336 12.1.4 Using the mean ergodic theorem. . . . . . . . . . . . . . . 339 12.1.5 The Amrein-Georgescu theorem. . . . . . . . . . . . . . . 340 12.1.6 Kato potentials. . . . . . . . . . . . . . . . . . . . . . . . 341 12.1.7 Applying the Kato-Rellich method. . . . . . . . . . . . . . 343 12.1.8 Using the inequality (12.7). . . . . . . . . . . . . . . . . . 344 12.1.9 Ruelle’s theorem. . . . . . . . . . . . . . . . . . . . . . . . 345 12.2 Non-negative operators and quadratic forms. . . . . . . . . . . . 345 12.2.1 Fractional powers of a non-negative self-adjoint operator. 345 12.2.2 Quadratic forms. . . . . . . . . . . . . . . . . . . . . . . . 346 12.2.3 Lower semi-continuous functions. . . . . . . . . . . . . . . 347 12.2.4 The main theorem about quadratic forms. . . . . . . . . . 348 12.2.5 Extensions and cores. . . . . . . . . . . . . . . . . . . . . 350 12.2.6 The Friedrichs extension. . . . . . . . . . . . . . . . . . . 350 12.3 Dirichlet boundary conditions. . . . . . . . . . . . . . . . . . . . 351 12.3.1 The Sobolev spaces W 1,2 (Ω) and W 1,2 0 (Ω). . . . . . . . . 352 [...]... 28 CHAPTER 1 THE TOPOLOGY OF METRIC SPACES As Q does not contain a constant term, and A is an algebra, Q(f 2 ) ∈ A for any f ∈ A Since we are assuming that |f | ≤ 1 we have Q(f 2 ) ∈ A, Q(f 2 ) − |f | and Since we are assuming that A is closed under completing the proof of the lemma · ∞ ∞ 0 there is a compact set C such that |f | < on the complement of C We can... y) and so using the fact that F = 1 and the triangle inequality gives |F (x2 ) − F (x1 )| ≤ x2 − x1 ≤ x2 + y + x1 + y This completes the proof of the proposition, and hence of the Hahn-Banach theorem over the real numbers We now deal with the complex case If B is a complex normed vector space, then it is also a real vector space, and the real and imaginary parts of a complex linear function are real. .. contains the constants and separates points, then A = CR (M) Proof The only A- level sets are points But since f − f (a) conclude that df (M) = 0, i.e f ∈ A for any f ∈ CR (M) QED 1.17 {a} = 0, we The Hahn-Banach theorem This says: Theorem 1.17.1 [Hahn-Banach] Let M be a subspace of a normed linear space B, and let F be a bounded linear function on M Then F can be extended so as to be defined on all of. .. exists an A level set satisfying(1.3) In fact, every f -minimal set is an A level set Proof Let E be an f -minimal set Suppose it is not an A level set This means that there is some h ∈ A which is not constant on A Replacing h by ah + c where a and c are constant, we may arrange that min h = 0 x∈E and max h = 1 x∈E Let 1 2 } and E1 : {x ∈ E| ≤ x ≤ 1} 3 3 These are non-empty closed proper subsets of E, and... open intervals (Equally well, we could use half open intervals of the form [a, b), for example.) It is clear that if A ⊂ B then m∗ (A) ≤ m∗ (B) since any cover of B by intervals is a cover of A Also, if Z is any set of measure zero, then m∗ (A ∪ Z) = m∗ (A) In particular, m∗ (Z) = 0 if Z has measure zero Also, if A = [a, b] is an interval, then we can cover it by itself, so m∗ ( [a, b]) ≤ b − a, and hence... STONE-WEIERSTRASS THEOREM 1.15 27 The Stone-Weierstrass theorem This is an important generalization of Weierstrass’s theorem which asserted that the polynomials are dense in the space of continuous functions on any compact interval, when we use the uniform topology We shall have many uses for this theorem An algebra A of (real valued) functions on a set S is said to separate points if for any p, q ∈ S,... said to be of first category if it is a countable union of nowhere dense sets Then Baire’s category theorem can be reformulated as saying that the complement of any set of first category in a complete metric space (or in any Baire space) is dense A property P of points of a Baire space is said to hold quasi - surely or quasi-everywhere if it holds on an intersection of countably many dense open sets . metric space X is called separable if it has a countable subset {x j } of points which are dense. For example R is separable because the rationals are countable and dense. Similarly, R n is separable. 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. is 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