# DEPARTMENT OF MATHEMATICS

## Analysis Syllabus

##### Measure Theory
1. 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.
2. Definition of the Lebesgue integral. Simple functions.
3. The main convergence theorems: Monotone, Fatou, Lebesgue dominated convergence theorem.
4. Distribution function. Chebychev inequality. Pushforward formula $f*\mu$.
5. Applications of Dominated convergence: differentiation under the integral sign, continuity of functions defined by integrals.
6. $L^1(\mathbb{R}^n,dm)$ and more generally $L^p(X,\mu)$. Basic density theorems: simple functions, $C_c(\mathbb{R}^n)$.
7. General measure spaces $X,\mathcal{M},\mu)$. Generalization of the simplest properties of Lebesgue measure. Carathéodory criterion for measurability. Integration. Fubini theorem.
8. Egorov and Lusin theorems.
9. 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.
10. Absolute continuity of $\nu$ with respect to $\mu$, $\varepsilon$-$\delta$ test. Uniformly integrable sequence. Vitali convergence theorem.
11. Fundamental theorem of Calculus 1: differentiating the integral. Lebesgue differentiation theorem on $\mathbb{R}^1$. Lebesgue differentiation theorem in $\mathbb{R}^n$.
12. 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
1. 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.
2. Linear functionals and the Riesz representation theorem. Application: Radon-Nikodym theorem.
3. Linear operators on a Hilbert space. Norm of an operator. Unitary or bounded self-adjoint operators. Schur-Young inequality.
4. $L^p$ spaces. Hölder 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.
5. 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.
6. 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 $\mathbb{R}^n$.
7. 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.
8. $C(X)$ for a compact Hausdor space. Compact sets in $C(X)$: Arzela-Ascoli theorem. Weierstrass density theorem and some of the Stone-Weierstrass theorem.
9. Weak and weak* topologies. Banach-Alaoglu theorem.
10. 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.