M.S. Level Material

M.S. Advanced Calculus Syllabus

Real Analysis

Lebesgue Measure and Integral

  • Definition and properties of Lebesgue measure and integral

  • Convergence theorems

  • Differentiation of functions of bounded variation, differentiation of an indefinite integral, absolute continuity

  • Holder and Minkowski inequalities, Lp spaces: completeness, representation of dual spaces

  • Product measures, Fubini and Tonelli theorems

Banach Spaces

  • Dual spaces, Hahn-Banach theorem,embedding in second dual space

  • Linear operators, closed graph and open mapping theorems, uniform boundedness principle

  • Hilbert spaces, orthonormal sets, Riesz-Fischer theorem, representation of bounded linear functionals

  • Function spaces, Ascoli's theorem, Stone-Weierstrass theorem

General Measure

  • Signed measures, Radon-Nikodym theorem

  • Outer measures, extension of a measure defined on an algebra or semialgebra

  • Riesz representation theorem for positive linear functionals on C(X), dual space of C(X)

References

Folland: Real Anaylsis
Royden: Real Analysis
Rudin: Real and Complex Analysis
Rudin: Functional Analysis
Lang: Analysis
Reisz & Sz-Nagy: Functional Analysis
Dieudonne: Foundations of Modern Analysis