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PG–0213 MMS–25 M.Sc. DEGREE EXAMINATION DECEMBER 2010. Second Year (AY – 2006-07 batch onwards) Mathematics TOPOLOGY AND FUNCTIONAL ANALYSIS Time : 3 hours Maximum marks : 75 SECTION A — (5 5 = 25 marks) Answer any FIVE questions. 1. Prove that intersection of two topologies on a set X is a topology on X. 2. Let Y be a subspace of X. If A is closed in Y and Y is closed in X, prove that A is closed in X. 3. Let A be a connected subset of X. If A B A prove that B is also connected. 4. Prove that a subspace of a Hausdorff space is Hausdorff. 5. Prove that the image of compact space under a continuous map is compact. 6. State and prove closed graph theorem. 7. State and prove Schwarz inequality. 8. If T is an operator on H for which (Tx , x ) 0 for all x , prove that T 0. SECTION B — (5 10 = 50 marks) Answer any FIVE questions. 9. Let ( X , T1 ) and (Y , T2 ) be two topological spaces. Let U V , where U is T1 open in X and V is T2 open in Y. Prove that is a basis for a topology on X Y. 10. Prove that every finite set in a Hausdorff space is closed. 11. Prove that a space X is connected if and only if the only subsets of X that are both open and closed in X are the empty set and X itself. 12. Suppose that X has a countable basis. Prove that there exists a countable dense subset of X. 13. State and prove Urysohn lemma. 14. Describe the Hahn–Banach theorem. 2 PG–0213 15. Prove that a closed convex subset C of a Hilbert space H contains a unique vector of smallest norm. 16. If T is an operator on H, prove that T is normal if and only if its real and imaginary parts commute. ——––––––––– 3 PG–0213