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राष्ट्रीय विज्ञान शिक्षा एवंअनुसंधान संस्थान
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  2. MA603 - Analysis I

MA603 - Analysis I

By admin_sms on Thu, 12/03/2015 - 21:35
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

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