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PAPER II: REAL ANALYSIS

UNIT I
:

RIEMANN STILTJES INTEGRAL: Definition and Existence of the Integral – properties of the integral – Integration and differentiation – Integration of vector valued function – rectifiable curves.

UNIT II:
Uniform convergence and continuity – uniform convergence and integration - uniform convergence and differentiation – equicontinuous families of functions – The Stone Weirstrass theorem

UNIT III:
FUNCTIONS OF SEVERAL VARIABLES: Linear transformation – contraction principle – Inverse function theorem – Implicit function theorem – determinants – derivatives of higher order – differentiation of integrals

UNIT IV:
LEBESGUE MEASURE AND LEBESGUE INTEGRAL: Outer measure – Measurable sets and Lebesgue measure – Measurable functions – Littlewood’s Theorem - The Lebesgue integral of bounded functions over a set of finite measure – integral of a non – negative function – General Lebesgue Integral – convergence in measure

UNIT V:
DIFFERENTIATION AND INTEGRATION Differentiation of monotone function – Differentiation of an Integral – Absolute continuity – The Minkovski and Holder Inequalities – Convergence and Completeness – Bounded linear functionals on the Lp spaces.

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