Syllabus for the Preliminary Examination in Analysis
The objective of this syllabus is to aid students in attaining a broad understanding of analysis techniques that are the basic stepping stones to contemporary research. The prelim exam normally consists of eight to ten problems, and the topics listed below should provide useful guidelines and strategy for their solution. It is assumed that students are familiar with the subject matter of the undergraduate analysis courses M365C and M361. The first part of the Prelim examination will cover Real Analysis. The second part of the prelim examination will cover Complex Analysis.
1. Measure Theory and the Lebesgue Integral
Basic properties of Lebesgue measure and the Lebesgue integral on Rn (see , Ch. 1-4) and general measure and integration theory in an abstract measure space (see , Ch. 11-12; and especially , Ch. 1-2). Lp spaces (see , Ch. 3); convergence almost everywhere, in norm and measure; approximation in Lp-norm and Lp-Lq duality; integration in product spaces (see , Ch. 8) and convolution on Rn; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.
2. Holomorphic Functions and Contour Integration
Basic properties of analytic functions of one complex variable (see , Ch. 4-5; , Ch. 4-7; , Ch. 4-8; or , Ch. 10-12 and 15). Integration over paths, the local and global forms of Cauchy's Theorem, winding number and residue theorem, harmonic functions, Schwarz's Lemma and the Maximum Modulus theorem, isolated singularites, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.
The relationship between differentiation and the Lebesgue integral on a real interval (see , Ch. 5), derivatives of measures (see , Ch. 5), absolutely continuous functions and absolute continuity between measures, functions of bounded variation.
4. Specific Important Theorems
Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Fubini-Tonelli theorems about product measures and integration on product spaces, Cauchy's theorem and integral formulas, Maximum Modulus theorem, Rouche's theorem, Residue theorem, and Fundamental Theorem of Calculus for Lebesgue Integrals. Students should be familiar with Minkowski's Inequality, Holder's Inequality, Jensen's Inequality, and Bessel's Inequality.
1. L. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1979.
2. J.B. Conway, Functions of One complex Variable, second edition, Springer-Verlag, New York, 1978.
3. G.B. Folland, Real Analysis, second edition, John Wiley, New York, 1999.
4. B. Palka, An Introduction to Complex Function Theory, second printing, Springer-Verlag, New York, 1995.
5. H.L. Royden, Real Analysis, Macmillan, New York, 1988.
6. W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, New York, 1987.
7. R. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977.
Syllabus for M365C -- Introduction To Analysis
The real number system and euclidean spaces: The axiomatic description of the real number system as the unique complete ordered field; the complex numbers; euclidean space R.
Metric spaces: Elementary metric space topology, with special emphasis on euclidean spaces; sequences in metric spaces --- limits, accumulation points, subsequences, etc.; Cauchy sequences and completeness; compactness in metric spaces; compact sets in R; connectedness in metric spaces; countable and
Continuity: Limits and continuity of mappings between metric spaces, with particular attention to real-valued functions defined on subsets of R; preservation
of compactness and connectedness under continuous mapping; uniform continuity.
Differentiation on the line: The definition and geometric significance of the derivative of a real-valued function of a real variable; the Mean Value Theorem and its consequences; Taylor's theorem; L'Hospital's rules.
Riemann integration on the line: The definition and elementary properties of the Riemann integral; existence theorems for Riemann integrals; the Fundamental Theorems of Calculus.
Sequences and series of functions: Uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation.
(An appropriate text might be Rudin's Principles of Mathematical Analysis, and the course should cover roughly its first seven chapters.)