Punjabi University 2008 M.Sc Mathematics SET TOPOLOGY AND FUNCTIONAL ANALYSIS - Question Paper

M.Sc Mathematics DEGREE EXAMINATION, 2008

(MATHEAMTICS)
(SECOND YEAR)
(PAPER-VI)
220. SET TOPOLOGY AND FUNCTIONAL ANALYSIS

(Revised Regulations)

(Including Lateral Entry)

2nd June) (Time: three Hours
Maximum: 100 Marks

PART-A (8×5=40)
ans any 8 ques.
All ques. carry equal marks

1. describe the open set in a metric space. If X is a metric space, prove that any union of open sets in X is open.

2. describe the terms:
a) Discrete Topology
b) Complete Metric space
c) Sub-base


3. Prove that any continuous image of a compact space is compact..

4. State Urysohn’s lemma.

5. describe the terms:
a) Connected space
b) Connected subspace
c) Disconnected space

6. Prove that the components of a totally disconnected space are its points.

7. State and prove the Schwarz inequality.

8. State the open mapping theorem.

9. provide an example of a Hilbert space.

10. Let N and N' be normed linear spaces and T a linear transformation of N into N'. Then prove the subsequent conditions on T are all equivalent to 1 a different.

PART-B (3×20=60)
ans any 3 ques.
All ques. carry equal marks

11. a) State and prove Cantor’s intersection theorem.

b) Let X be a topological space and A an arbitrary subset of X. Prove that ={x : every neighbourhood of x intersects A}.

12. a) State and prove Tychonoff’s theorem.

b) Let X and Y be metric spaces and f a mapping of X into Y. Then prove that f is continuous if and only if, f -1 (G) is open in X whenever G is open in Y.

13. Prove that a subspace of the real line R is connected if any only if, it is an interval. In particular, show that R is connected.

14. State and prove Hahn-Banach theorem.

15. Prove that every non-zero Hilbert space contains a complete orthonormal set.

Earning:   Approval pending.