Tamil Nadu Open University (TNOU) 2009-2nd Year M.Sc Mathematics Tamilnadu open university Maths Topology and functional analysis - Question Paper

M.Sc. DEGREE EXAMINATION JUNE 2009.

(AY 2006-2007 batch onwards)

Second Year

Mathematics

TOPOLOGY AND FUNCTIONAL ANALYSIS

Time : 3 hours Maximum marks : 75

PART A (5 5 = 25 marks)

Answer any FIVE questions.

1.         Let B,B' the bases for topologies respectively. Show that is finer than , if and only if for each x and each B containing x there is a B' such that

2.         Let be a family of spaces and Show that in both product and box topologies

3.         Show that finite product of connected spaces is connected.

4.         State and prove extreme value theorem.

5.         Show that subspace of a regular space is regular.

6.         Let be a linear transformation from a normed linear spaced to another. Show that the following conditions are equivalent.

             (a) T is continuous

             (b) T is continuous at origin

             (c) There exist a real number K > 0 such that

7.         State and prove uniform bondedness theorem.

8.         If T is an operator on a Hilbert Space H such that (T (x), x) = 0 for all x, show that T = 0.

PART B (5 10 = 50 marks)

Answer any FIVE questions.

9.         Show that the topologies on in duced by the Euclidean metric and the square metric are the same as the product topology.

10.       (a) Let where x is a metric space and y is a topological space. Show that f is continuous if and only if implies

             (b) Let A X and A' be the set of all limit points of A. Show that

11.       (a) Show that the ordered square with ordered topology is connected but not path connected.

             (b) If X is locally path connected show that the components and the path components of X are the same.

12.       Let X be a locally compact Hausdorff space. Show that there exists a space Y such that X is a subspace of Y, Y - X consists of a single point and Y is compact Hausdorff.

13.       State and prove Uryshons lemma.

14.       State and Prove Hahn Banach theorem.

15.       Let H be a Hilbert space and f be an arbilrary functional. Show that there exists a unique such that f (x) = (x, y).

16.       (a) Show that an operator T is unitary if and only if it is an isometric isomorphism.

             (b) Show that an operator T is normal if and only if its real and imaginary parts commute.