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Analysis notes Aidan Backus December 2019 Contents I Preliminaries2 1 Functional analysis3 1.1 Locally convex spaces...............................3 1.2 Hilbert spaces...................................6 1.3 Bochner integration................................6 1.4 Duality....................................... 11 1.5 Vector lattices................................... 13 1.6 Positive Radon measures............................. 14 1.7 Baire categories.................................. 17 2 Complex analysis 20 2.1 Cauchy-Green formula.............................. 20 2.2 Conformal mappings............................... 22 2.3 Approximation by polynomials.......................... 24 2.4 Sheaves...................................... 26 2.5 Subharmonicity.................................. 28 2.6 Operator theory.................................. 30 II Dynamical systems 31 3 Elementary dynamical systems 32 3.1 Types of dynamical systems........................... 32 3.2 Properties of the irrational rotation....................... 34 4 Ergodic theory 37 4.1 The mean ergodic theorem............................ 37 4.2 Pointwise ergodic theorem............................ 41 4.3 Ergodic systems.................................. 45 4.4 Properties of ergodic transformations...................... 47 4.5 Mixing transformations.............................. 48 4.6 The Hopf argument................................ 54 1 5 Flows on manifolds 59 5.1 Ergodic theorems for flows............................ 59 5.2 Geodesic flows in hyperbolic space........................ 60 6 Topological dynamics 68 6.1 Recurrence in topological dynamics....................... 68 7 Completely integrable systems 71 7.1 Symplectic geometry............................... 71 7.2 Hamiltonian flows on symplectic spaces..................... 73 7.3 Hamiltonian flows on symplectic manifolds................... 75 7.4 Complete integrability.............................. 76 7.5 The Liouville-Arnold-Jost theorem....................... 77 7.6 The Toda lattice................................. 81 7.7 The QR algorithm................................ 88 8 Complex dynamics 91 8.1 Siegel's KAM theorem.............................. 91 9 Entropy and information theory 96 9.1 Shannon's axioms of entropy........................... 96 9.2 Conditional entropy................................ 97 9.3 The entropy of a group action.......................... 98 9.4 Topological entropy................................ 101 III Operator algebras 106 10 Banach algebras 107 10.1 The spectrum................................... 107 10.2 Ideals....................................... 109 10.3 The holomorphic functional calculus...................... 110 11 C∗-algebras 111 11.1 Weights...................................... 111 11.2 The GNS construction.............................. 112 11.3 The Bp spaces................................... 114 11.4 Representation theory of groups......................... 115 11.5 Compact operators................................ 116 12 Generators and relations 121 12.1 Construction of maximally free algebras.................... 121 12.2 Tensor products of C∗-algebras......................... 122 2 13 Representation theory of locally compact groups 124 13.1 Noncommutative dynamical systems...................... 124 13.2 Group actions on locally compact spaces.................... 129 13.3 Semidirect products of groups.......................... 134 13.4 The Heisenberg commutation relations..................... 135 13.5 Projective representations............................ 136 14 Noncommutative geometry 140 14.1 Quantum tori................................... 140 14.2 The 2-torus.................................... 140 14.3 Noncommutative smooth manifolds....................... 142 14.4 Noncommutative vector bundles......................... 144 IV Complex analysis 147 15 Holomorphy in several complex variables 148 15.1 Cauchy-Riemann equations............................ 148 15.2 Basic properties.................................. 150 15.3 Plurisubharmonicity and domains of holomorphy............... 152 15.4 Hormander L2 estimates............................. 161 16 Multivariable holomorphic functional calculus 164 16.1 The Gelfand transform.............................. 164 16.2 The holomorphic functional calculus...................... 168 17 Algebraic geometry 170 17.1 Schemes and varieties............................... 170 17.2 Formal power series................................ 171 17.3 Bergman kernels................................. 172 17.4 Quillen's theorem................................. 173 18 Line bundles over complex varieties 179 18.1 Holomorphic line bundles............................. 180 18.2 Integration on Hermitian line bundles...................... 183 18.3 Asymptotics for the Bergman kernel...................... 184 18.4 Morphisms into projective space......................... 193 18.5 Zworski's conjecture............................... 195 V Harmonic analysis 196 19 Rearrangement-invariant spaces 197 19.1 Log-convexity for Lp norms........................... 198 19.2 Lorentz norms................................... 199 3 20 Decoupling theory 202 20.1 Square-root cancellation............................. 202 20.2 Decoupling estimates............................... 204 21 Pseudodifferential calculus 211 21.1 The Kohn-Nirenberg quantization........................ 211 21.2 The Calderon-Vallaincourt theorem....................... 214 21.3 Composition estimates.............................. 219 21.4 The Gabor transform............................... 221 VI General relativity 222 22 Lorentzian geometry 223 22.1 Axioms of special relativity........................... 223 22.2 Lorentz transformations............................. 223 22.3 Riemannian geometry............................... 224 22.4 Causality..................................... 226 23 The Cauchy problem in general relativity 227 23.1 The Einstein equation.............................. 227 23.2 Initial-data sets.................................. 229 23.3 Well-posedness for the vacuum equation.................... 230 24 General relativity in spherical symmetry 233 24.1 Double-null pairs................................. 233 24.2 Local rigidity................................... 235 24.3 Einstein-Maxwell equations........................... 237 25 Cosmic censorship 241 25.1 Einstein-Maxwell-charged scalar field equations................ 242 25.2 The structure of toy models........................... 244 25.3 Penrose inequalities................................ 246 25.4 Recent progress on strong censorship...................... 248 4 Part I Preliminaries 5 Chapter 1 Functional analysis Here we treat functional analysis in a high level of abstraction. Throughout these notes, we mean by f g that there is a universal constant C > 0 such that f ≤ Cg. 1.1 Locally convex spaces Fix a vector space V . Definition 1.1. V is said to be a topological vector space if it is equipped with a topology for which addition and multiplication are continuous. Definition 1.2. V is said to be locally convex if V is equipped with a family of seminorms Pα and the initial topology with respect to the Pα. This is the smallest topology containing the open sets Pα([0;")) for each α and each " > 0 and which is translation-invariant. The most useful examples of locally convex spaces are Banach spaces. Definition 1.3. V is said to be a Banach space if V is equipped with the topology arising from a complete norm. Definition 1.4. If V is a topological vector space, then the dual space of V ∗ is the space of continuous linear maps V ! C. Definition 1.5. Let W be a Banach space and define a norm on Hom(V; W ) by jjT jj = sup jjT vjj: jjv||≤1 So V ∗ is a normed space, V ∗ ⊆ Hom(V; C). In general it is very difficult to construct elements of V ∗. In general we cannot guarantee constructively that V ∗ is nontrivial. On the other hand, it is often impossible to construct linear functions which are discontinuous (for example, any linear functional on a Banach space will be continuous if it was constructed without the axiom of choice). 6 Definition 1.6. A function f : V ! C is said to be sublinear if it obeys the triangle inequality and if for each c > 0 and x 2 V , f(cx) = cf(x). Obviously seminorms are sublinear. Minkowski gauges are another useful example. Definition 1.7. Let K ⊆ V . Then: 1. K is convex if for each x; y 2 K, c 2 [0; 1], cx + (1 − c)y 2 K. 2. K is balanced if for each c 2 [0; 1], cK ⊆ K. If K is balanced and convex, then the Minkowski gauge of K is the functional pK (x) = inf c: cK3x Notice that the balanced condition suggests that K needs to be close to the origin. Moreover, Minkowski gauges are sublinear. Sublinear estimates allow us to construct functionals using the axiom of choice, while still guaranteeing that they are continuous. Theorem 1.8 (Hanh-Banach). Assume that p : V ! C is sublinear, U ⊂ V a subspace, and f : U ! C a linear functional. If f is dominated by p, i.e. for each x 2 U, jf(x)j ≤ jp(x)j, then f extends to V . In general the extension of f will only be unique in case U is dense. So we have to use the axiom of choice to construct f. Proof. The extension to the complex case is trivial so we replace C with R. Assume that f is defined on a space W , U ⊆ W ⊂ V . Choose v 2 V n W and define f(v) such that for each w 2 W and s; t ≥ 0, p(w − sv) p(w + tv) − f(w) ≤ h(v) ≤ : s t This is always possible because f((t + s)w) ≤ p((t + s)w) = p((t + s)w + tsv − tsv) ≤ p(sw + stv) + p(tw − stv) so f(w) − p(w − sv) p(w + tv) −
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