MTH605: Topics in analysis

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• Lebesgue integral and construction of Lebesgue measure on ${\mathbb{R}}^n$, Non measurable sets, Fatou's Lemma and Convergence Theorems, $L^p$ spaces and their completeness, Reisz representation Theorem for $C_0(X)$ ; $X$ locally compact.
• Complex Measure, Radon-Nikodym Theorem, Differentiation of an integral, Absolutely continuous of functions, Hahn decomposition Theorem, Product measure and Fubini Theorem.
• Fourier series and Fourier transform, $L^2$ theory.
• Theory of Distributions.

Recommended Reading

• Walter Rudin, Real and Complex Analysis, McGraw-Hill International Editions, Mathematics Series, McGraw-Hill Education (1987).
• Walter Rudin, Functional Analysis, Tata McGraw-Hill (1990). (for the Theory of Distributions)
• Halsey L. Royden, Real Analysis, Prentice Hall, Third Edition (1988).