RA2 (MSC-Math 2nd)

UNIT-I

(i) Differentiation: Differentiation of vector-valued functions.

(ii) Functions of several variables: The space of linear transformations on Rn to Rm as a metric space. Differentiation of a vector-valued function of several variables. The inverse function theorem The implicit function theorem.

(iii) Lebesgue measure: Introduction. Outer measure. Measurable sets and Lebesgue measure. A nonmeasurable set. Measurable functions. Littlewood's three principles.

UNIT-II

(iv) The Lebesgue integral: The Lebesgue integral of a bounded function over a set of finite measure. The integral of a non-negative function. The general Lebesgue integral. Convergence in measure.

(v) Differentiation and Integration: Differentiation of monotone functions. Differentiation of anintegral. Absolute continuity. Convex functions.