Analysis Examination Syllabus

Topics

Measure and Integration in n-space

Lebesgue measure

Lebesgue integral

Differentiation

Abstract Measure and Integration

Measures and Outer Measures

Extension and Completion theorems

Measurable Functions

Integration

Convergence

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

Bibliography

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.