Amrita Vishwa Vidyapeetham 2007 B.Sc Mathematics Real Part Analysis - Question Paper

M.Sc. DECEMBER 2007.
Real Part Analysis
Time : 3 hours Maximum : 100 marks
ans any 5 ques..
All ques. carry equal marks.
1. (a) Show that a set E is open if and only if its complement is closed.
(b) Show that compact subsets of metric spaces are closed.
(c) Show that every K-cell is compact.
2. (a) Show that every perfect set in is uncountable.
(b) Show that a subset E of is connected if and only if and implies
3. (a) State and prove the mean value theorem.
(b) Suppose f is a continuous mapping of into and f is differentiable in (a, b). Prove that there exists such that .
4. (a) Define Riemann-Stieltjes integration.
(b) If f is continuous on , prove that on .
(c) Assume increases monotonically and on . Let f be a bounded real function on . Prove that if and only if and

5. (a) State and prove Cauchy criterion for uniform convergence.
(b) Suppose uniformly on a set E in a metric space. Let x be a limit point of E and suppose that . Prove that converges, and .
6. (a) Suppose is a sequence of functions, differentiable on and converges for a few point on . If converges uniformly on , then prove that converges uniformly on to a function f, and
.
(b) State and prove the Weierstrass approximation theorem.
7. (a) State and prove the inverse function theorem.
(b) State and prove the Gauss Divergence theorem.
8. (a) Prove that the outer measure of an interval is its length.
(b) State and prove monotone convergence theorem

Earning:   Approval pending.