220 SET TOPOLOGY AND FUNCTIONAL ANALYSIS Punjabi University 2008 Question Paper
Punjabi University Question Paper
M.Sc Mathematics DEGREE EXAMINATION, 2008
MATHEMATICS
Second Year
Paper - VI
220 SET TOPOLOGY AND FUNCTIONAL ANALYSIS
(Revised Regulations)
(Including Lateral Entry)
2nd June) (Time: 3 Hours
Maximum: 100 Marks
PART-A (8×5=40)
Answer any EIGHT questions
All questions carry equal marks
1. Define 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. Define 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. Define 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. Give 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 following conditions on T are all equivalent to one another.
PART-B (3×20=60)
Answer any THREE questions
All questions 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 : each 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.
Punjabi University Question Paper
M.Sc Mathematics DEGREE EXAMINATION, 2008
MATHEMATICS
Second Year
Paper - VI
220 SET TOPOLOGY AND FUNCTIONAL ANALYSIS
(Revised Regulations)
(Including Lateral Entry)
2nd June) (Time: 3 Hours
Maximum: 100 Marks
PART-A (8×5=40)
Answer any EIGHT questions
All questions carry equal marks
1. Define 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. Define 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. Define 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. Give 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 following conditions on T are all equivalent to one another.
PART-B (3×20=60)
Answer any THREE questions
All questions 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 : each 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.
0 comments:
Pen down your valuable important comments below