Doctorate Real Analysis Syllabus

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)


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