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Analysis Examination Syllabus


Measure and Integration in n-space

Lebesgue measure

Lebesgue integral


Abstract Measure and Integration

Measures and Outer Measures

Extension and Completion theorems

Measurable Functions



Jordan Decomposition

Lebesgue Decomposition

Radon-Nikodym theorem

Product measures and integrals

Fubini theorem

L^p Spaces

Functional Analysis

Elementary properties of normed linear spaces and linear operators

Hahn-Banach Theorem

Open Mapping Theorem

Principle of Uniform Boundedness

Closed Graph Theorem


  1. Royden, H.L., Real Analysis, 1968, Chapter 3-6, 10-13
  2. Rudin, W., Real and Complex Analysis, 1966, Chapters 1-3; 5-8
  3. Segal, I.E. & Kunze, R.A., Integrals and Operators, 2nd Ed. 1978, Chapters II-IV
  4. Dunford, N. & Schwartz, J.T., Linear Operators, Part I, 1958, Chapter III
  5. Hewitt, E. & Stromberg, K., Real Abstract Analysis, 1969
  6. Saks, S., Theory of the Integral, 1937.