Analysis Syllabus
Measure Theory
- Definition of Lebesgue measure \((\mathbb{R}^n,\mathcal{M},m)\). Elementary sets, Lebesgue measurable sets and measurable functions. Outer measure. Borel measurability, continuous and semi-continuous functions. Basic properties of Lebesgue measures: monotonicity, regularity.
- Definition of the Lebesgue integral. Simple functions.
- The main convergence theorems: Monotone, Fatou, Lebesgue dominated convergence theorem.
- Distribution function. Chebychev inequality. Pushforward formula \(f*\mu\).
- Applications of Dominated convergence: differentiation under the integral sign, continuity of functions defined by integrals.
- \(L^1(\mathbb{R}^n,dm)\) and more generally \(L^p(X,\mu)\). Basic density theorems: simple functions, \(C_c(\mathbb{R}^n)\).
- General measure spaces \(X,\mathcal{M},\mu)\). Generalization of the simplest properties of Lebesgue measure. Carathéodory criterion for measurability. Integration. Fubini theorem.
- Egorov and Lusin theorems.
- Modes of convergence (pointwise a.e., \(L^1\), in measure, uniform, almost uniform. Escape to horizontal infinity, escape to width infinity, escape to vertical infinity. Typewriter sequence.
- Absolute continuity of \(\nu\) with respect to \(\mu\), \(\varepsilon\)-\(\delta\) test. Uniformly integrable sequence. Vitali convergence theorem.
- Fundamental theorem of Calculus 1: differentiating the integral. Lebesgue differentiation theorem on \(\mathbb{R}^1\). Lebesgue differentiation theorem in \(\mathbb{R}^n\).
- Fundamental theorem of Calculus 2: A.e. differentiability of monotone functions. Functions of bounded variation. A.e. differentiability of BV functions. Absolutely continuous functions. Integrating the derivative.
References
[F1] G. B. Folland, Real analysis. Modern techniques and their applications. Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New
York, 1999.
[Roy] H. L. Royden, Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.
[R] W. Rudin, Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987.
[S] E. M. Stein and R. Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ, 2005
[Tao1] T. Tao, An introduction to measure theory. Graduate Studies in Mathematics, 126. American Mathematical Society, Providence, RI, 2011.
Functional Analysis
- Hilbert space (separable). Orthonormal basis. Parseval identity. Closed subspaces and orthogonal projection. \(L^2(X,\mu)\). Completeness and density theorems. \(L^2(S^1)\). Fourier series. Parseval theorem, Bessel inequality.
- Linear functionals and the Riesz representation theorem. Application: Radon-Nikodym theorem.
- Linear operators on a Hilbert space. Norm of an operator. Unitary or bounded self-adjoint operators. Schur-Young inequality.
- \(L^p\) spaces. Hölder and Minkowski inequalities. Minkowski inequality for integrals. Duality theorem. Weak convergence in \(L^p\). \(L^p(\mathbb{R}^n,dx)\). Convolution. Approximate identities and density theorems.
- Banach spaces. Hahn-Banach theorem. Baire Category and its application to bounded operators \(T : X\to Y\) between Banach spaces. Open Mapping and Closed Graph theorems. Uniform boundedness principle. Applications: Convergence of Fourier series. Decay of Fourier coefficients.
- Compact sets in Hilbert space. Compact and Hilbert Schmidt operators. Spectral theorem for compact self-adjoint operators. Hilbert space method for solving constant coefficient PDE's on bounded domains in \(\mathbb{R}^n\).
- Fourier transform as a unitary operator on \(L^p(\mathbb{R}^n,dx)\). Plancherel and inversion formulae. Schwartz class functions. Fourier conjugation of differentiation and multiplication.
- \(C(X)\) for a compact Hausdor space. Compact sets in \(C(X)\): Arzela-Ascoli theorem. Weierstrass density theorem and some of the Stone-Weierstrass theorem.
- Weak and weak* topologies. Banach-Alaoglu theorem.
- Radon measures: Positive linear functionals. Dual space of \(C_0(X)\).
References
[E] L. C. Evans, Appendix to Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
[F1] G. B. Folland, Real analysis. Modern techniques and their applications. Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.
[Roy] H. L. Royden, Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.
[R] W. Rudin, Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987.
[S] E. M. Stein and R. Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ, 2005
[S2] E. M. Stein and R. Shakarchi, Functional Analysis. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ, 2005.
[Tao2] T. Tao, An epsilon of room, I: real analysis. Pages from year three of a mathematical blog. Graduate Studies in Mathematics, 117. American Mathematical Society, Providence, RI, 2010.
Complex Analysis
- Cauchy integral formula. Residue theorem. Calculation of integrals by residue methods. Mean value formula for harmonic functions.
- Isolated singularities of holomorphic functions. Taylor and Laurent expansions. Poles and essential singularities. Riemann's removable singularities theorem.
- Growth of holomorphic functions. Maximum modulus principle. Cauchy estimates. Liouville theorem and its generalizations (e.g. analytic functions of polynomial growth). Open mapping theorem for holomorphic functions.
- Zeros of analytic function. Argument principle. Rouche's theorem. Jensen and Poisson-Jensen formulae. Zeros of polynomials.
- Normal families and uniform convergence on compact sets. Montel's theorem. Hurwitz's theorem. Compactness for uniformly bounded families of holomorphic functions.
- Holomorphic functions as conformal maps. Automorphisms of the plane, upper halfplane, disc, punctured disc, annulus. Schwarz Lemma and Schwarz-Pick Lemma.
- Riemann mapping theorem for simply connected domains $\neq \mathbb{C}$.
Harmonic and subharmonic functions. Dirichlet boundary problem. Perron's method. Green's function and the Riemann mapping theorem.
References
[1] T. W. Gamelin, Complex Analysis. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2001.[2] R.E. Greene and S. G. Krantz, Function theory of one complex variable. Third edition. Graduate Studies in Mathematics, 40. American Mathematical Society, Providence, RI, 2006.
[3] E.M. Stein and R. Shakarchi, Complex analysis. Princeton Lectures in Analysis, II. Princeton University Press, Princeton, NJ, 2003.