## Syllabus Topics

This exam will test your working knowledge of basic real, complex and functional analysis. You will be required to demonstrate an ability to use standard results and techniques to solve problems, including special cases of standard theorems which do not require long arguments.

We will not emphasize the memorization of statements of theorems nor of long proofs of standard theorems.

The student is urged to work on the problems in the relevant sections and chapters in the reference books.

The syllabus is divided into the topics of Complex Analysis and Real and Functional Analysis.

### Complex Analysis

1. Definition of Holomorphic fuctions with examples, including logarithms, roots, and Möbius transformations.

2. Cauchy-Riemann Equations.

3. Power Series Expansion and applications including the Identity Theorem.

4. Cauchy Integral Theorem.

5. Applications of Cauchy Integral Theorem to evaluating Riemann integrals and summation of infinite series, Rouché's Theorem, The Argument Principle, Open Mapping Theorem, Liouville's Theorem, The Fundamental Theorem of Algebra.

6. The Maximum Modulus Theorem and Applications including the Schwarz Lemma.

7. Limit properties of Holomorphic functions including Hurwitz' Theorem.

**References**

[1] Ahlfors,Complex Analysis, Chapters 2,3,4,6.

[2] Conway, Functions of One Complex Variable, Chapters 1,3,4,5,6.

[3] Schaum's Outline of Complex Variables, Chapters 3-8.

### Real and Functional Analysis

**A. Real Analysis on the Real Line**

σ-algebra, Borel sets, construction of Lebesgue measure, measurable sets, how to approximate a Lebesgue measurable set with positive measure from outside and inside, Cantor set, measurable functions, f(g(x)) is measurable if f is continuous and g is measurable.

Convergence a.e., convergence in measure, convergence in the mean and how they are related to each other, Egorov's Theorem, Luzin's Theorem.

Lebesgue integrals, Fatou's Lemma, Monotone Convergence Theorem, Lebesgue Dominated Convergence Theorem, differentiation across the integral sign; comparison of Riemann and Lebesgue integrals; sequence of functions with equi-absolutely continuous integrals: Vitali's Theorem on a set with finite measure ([1, pp 151-159], [4, pp 143, Exercise 10], equi-absolutely continuous integrals = uniformly continuous integrals).

Set of discontinuous points and differentiabilty of a monotone function, functions of bounded variation, absolutely continuous functions, fundamental theorem of calculus and its counter-example (Cantor's function). ([1, pp 204-220], [3, Chapt. 5].) Lebesgue's density theorem.

Lp spaces with 1≤p≤∞, Hölder inequality, Minkowski inequality, Jensen's inequality, convergence in Lp, completeness of spaces, dense sets of Lp( a,b) (1≤p<∞ )∶ the set of continuous functions, the set of polynomials, the set of step functions, orthonormal systems in L2(a,b), Bessel's inequality, completeness of orthonormal systems, Parseval's identity, Fourier trigonometric series of a funtion in L2(−π,π), Riesz Representation for bounded linear functionals on Lp (1≤p<∞). ([3, Chapt. 6], [1, Chapt. VII].)

**B. Abstract measure theory**

Abstract measure spaces, completion of measure spaces, one measure is orthogonal or absolute continuous with respect to another measure, Lebesgue decomposition, Radon-Nikodyn theorem; product measures, Fubini-Toneli Theorem.

**C. Functional Analysis**

Space of continuous functions, Weierstrass Theorem, Arzela-Ascoli Theorem ([3, Chapt. 9, Sec. 6, 7]).

Contraction Mapping Theorem on complete metric spaces with applications to initial value problems of ODE and integral equations ([2, Sec. 8], [5, Sec. 3.8]).

Hilbert spaces, Schwarz's inequality, orthogonal projection of a point onto a closed subspace, Riesz Representation Theorem ([4, pp 79-85], [5, Secs 6.1 and 6.2]). Bessel's inequality, Parseval's identity, Gram-Schmidt procedure, completeness of orthonormal sets, least-square approximation.

Normed vector spaces, Banach spaces, bounded operators, standard examples: C(K), Lp, little ℓp, etc. Hahn-Banach theorem, reflexive Banach spaces, open mapping theorem, closed graph theorem, weak convergence, strong convergence, weakly compact, strongly compact, the unit ball in an infinite dimensional normed vector space is not strongly compact.

**References**

[1] Natanson, Theory of Functions of a Real Variable.

[2] Kolmogorov and Fomin, Introductory Real Analysis.

[3] Royden, Real Analysis.

[4] Rudin, Real and Complex Analysis, 2nd edition.

[5] Friedman, Foundations of Modern Analysis.

[6] Folland, Real Analysis.

[7] Wheeden and Zygmund, Measure and Integral, an introduction to real analysis.

[8] Stein and Sharkarchi, Real Analysis, Measure Theory, Integration and Hilbert Spaces.

[9] Stein and Sharkarchi, Fourier Analysis.