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, HahnBanach theorem,embedding in second dual space

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

Hilbert spaces, orthonormal sets, RieszFischer theorem, representation of bounded linear functionals

Function spaces, Ascoli's theorem, StoneWeierstrass theorem
General Measure

Signed measures, RadonNikodym 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 & SzNagy: Functional Analysis
Dieudonne: Foundations of Modern Analysis