M603 - Analysis I

Course No: 
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, measurable functions 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's thoerem (without proof) and its applications.
$L^p$-spaces: Holder and Minkowski inequalities, completeness, Convolutions, Approximation by smooth functions.
Fourier analysis: Fourier Transform, Inverse Fourier transform, Plancherel Theorem for $\mathbb R$.
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.

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School of Mathematical Sciences

NISERPO- Bhimpur-PadanpurVia- Jatni, District- Khurda, Odisha, India, PIN- 752050

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