# Course

## MA603 - Analysis I

Course No:
MA603
Credit:
8
Approval:
2018
PG-Core
Syllabus:

Spaces of functions: Continuous functions on locally compact spaces, Stone-Weierstrass theorems, Ascoli-Arzela Theorem. Review of Measure theory: Sigma-algebras,measures, construction and properties of the Lebesgue measure, non-measurable sets, measurablefunctions and their properties. Integration: Lebesgue Integration, various limit theorems,comparison with the Riemann Integral, Functions of bounded variation and absolute continuity.Measure spaces: Signed-measures, Radon-Nikodym theorem, Product spaces, Fubini'sthoerem (without proof) and its applications. Lp-spaces: Holder and Minkowski inequalities,completeness, Convolutions, Approximation by smooth functions. Fourier analysis: FourierTransform, Inverse Fourier transform, Plancherel Theorem for Real numbers.

Reference Books:
1. D. S. Bridges, Foundations of Real and Abstract Analysis, GTM series, Springer Verlag 1997.
2. G. B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Wiley-Interscience/John Wiley Sons, Inc., 1999.
3. P. R. Halmos, Measure Theory, Springer-Verlag, 1974.
4. H. L. Royden, Real Analysis, Macmillan 1988.
5. W. Rudin, Real and Complex Analysis, TMH Edition, Second Edition, New-York, 1962.
6. Elliott H. Lieb and Michael Loss , Analysis, American Mathematical Society, 2001