### 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

• 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