Ranchi University PG Syllabus

Hi I want the post graduate syllabus in mathematics for the Ranchi University do can you please help me in the same as I need to make notes???

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Ranchi University is a university in Jharkhand state of India

Post Graduate Syllabus in Mathematics for the Ranchi University

DETAILED SYLLABUS SEMESTER-I Paper – MCG101 (Functional Analysis-I & Real Analysis-I ) Unit-1 Functional Analysis-I Total Lectures : 40 (Marks – 30) Baire category theorem. Normed linear spaces, continuity of norm function, Banach spaces, Spaces Cn, C [a,b] (with supmetric) , c0, lp (1 ≤ p ≤ ∞) etc; (10L) Linear operator, boundedness and continuity, examples of bounded and unbounded linear operators. (10L) Banach contracton Principle – application to Picard’s existence theorem and Implicit function theorem. (8L) Inner product, Hilbert spaces, examples such as l2 spaces, L2[a,b] etc; C-S inequality, Parallelogram law, Pythagorean law, Minkowski inequality, continuity and derivatives of functions from Rm to Rn . (12L) 6

Unit-2 Real Analysis-I Total Lectures : 25 (Marks – 20) Monotone functions and their discontinuities, Functions of bounded variation on an interval, their properties, Riemann-Stieltjes integral, existence, convergence problem and other properties. (12L) Lebesgue outer measure, countable subadditivity, measurable sets and their properties, Lebesgue measure, measurable functions, equivalent functions, continuity and measurability, monotonocity and measurability, operation on collection of measurable functions, pointwise limit of a sequence of measurable functions, measurability of Supremum and Infimum, simple function and measurable function. (13L)

Paper – MCG102 (Linear Algebra & Modern Algebra-I) Unit-1 Linear Algebra Total Lectures : 40 (Marks – 30) Vector spaces, Euclidean space, Unitary space, orthonormal basis, Gram-Schmidt orthogonalization process. (8L) Linear transformation in finite dimensional spaces, matrix of linear, rank and nullity, annihilator of a subset of a vector space. (5L) Eigen vectors, spaces spanned by eigen vectors, similar and congruent matrices, characteristic polynomial, minimal polynomial, diagonalization, diagonalization of symmetric and Hermitian matrices, Cayley-Hamilton theorem, reduction of a matrix to normal form, Jordan Canonical form. (17L) Quadratic form, Reduction to normal form, Sylvester’s law of inertia, simultaneous reduction of two quadratic forms, applications to Geometry & Mechanics. (10L)








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Near Shaheed Chowk, Morabadi,

Ranchi, Jharkhand 834001

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