**MTH 603: Real Analysis (4)**

*Several variable calculus:* A quick overview, the contraction mapping theorem, the inverse function theorem, the implicit function theorem.

Riemann integration in **R ^{n}**, n≥1.

*Lebesgue measure and integration:* Measures, measurable functions, integration of nonnegative and complex functions, modes of convergence, convergence theorems, product measure, Fubini's theorem, convolution, integration in polar coordinates.

Signed measures and differentiation, complex measures, total variation, absolute continuity, Fundamental theorem of calculus for Lebesgue integral, the Radon-Nikodym theorem and consequences.

**L ^{p}** spaces, the Hölder and Minkowski inequalities, Jensen's inequality, completeness, the Riesz representation theorem, dual of

**L**spaces.

^{p}*Suggested Books*:

- G.B. Folland, Real analysis: Modern techniques and their applications, 2nd Edition, Wiley.
- W. Rudin, Principles of Mathematical Analysis, 3rd Edition, Tata McGraw-Hill.
- W. Rudin, Real and Complex Analysis, 3rd Edition, Tata McGraw-Hill.
- E.M. Stein and R. Shakarchi, Functional Analysis: Introduction to further topics in analysis, Princeton lectures in analysis.
- T. Tao, Analysis I and II, 2nd Edition, TRIM Series 37, 38, Hindustan Book Agency.

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