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
