Tamil Nadu Open University (TNOU) 2009-1st Year M.Sc Mathematics Tamilnadu open university Maths Real analysis - Question Paper
M.Sc. DEGREE exam —
JUNE, 2009.
(AY 2006–07 batches onwards)
First Year
Mathematics
REAL ANALYSIS
Time : three hours Maximum marks : 75
PART A — (5 x five = 25 marks)
ans any 5 ques..
1. Show that a set E is open if and only if its complement is closed.
2. Show that the subsequential limits of a sequence in a metric space X form a closed subset of X.
3. Show that composition of 2 continuous functions is continuous.
4. Suppose f is continuous on , exists at a few point , g is described on an interval I which contains the range of f and g is differentiable at the point . If for , show that
h is differentiable at x.
5. Show that on if and only if for every there exists a partition p such that .
6. State and prove Wierstrass test for uniform convergence.
7. Show that a linear operator A on a finite dimensional vector space X is one-to-one if and only if the range of A is all of X.
8. Show that a linear operator A on is invertible if and only if .
PART B — (5 × 10 = 50 marks)
ans any 5 ques..
9. Show that every nonempty perfect set in is uncountable.
10. Let be a series of real numbers which converges, but not absolutely. Let . Show that there exists a rearrangement with partial sums such that
; .
11. Let E be a non compact set in . Show that
(a) There exists a continuous function on E which is not bounded.
(b) There is a continuous and bounded function on E which has no maximum.
If E is bounded show that there exists a continuous function on E which is not uniformly continuous.
12. State and prove L’Hospital’s rule.
13. Let f be a bounded function on and increases monotonically such that on . Show that if and only if and .
14. Let be a sequence of functions differentiable on such that converges for a few point . If converges uniformly on show that converges uniformly on to a function
f and .
15. State and prove Stone Weirstrass theorem.
16. State and prove Implicit function theorem.
——–––––––––
M.Sc. DEGREE EXAMINATION
JUNE, 2009.
(AY 200607 batches onwards)
First Year
Mathematics
REAL ANALYSIS
Time : 3 hours Maximum marks : 75
PART A (5 5 = 25 marks)
Answer any FIVE questions.
1. Show that a set E is open if and only if its complement is closed.
2.
Show that the subsequential limits of a sequence 
in a metric space X
form a closed subset of X.
3. Show that composition of two continuous functions is continuous.
4.
Suppose f is continuous on 
,
exists at some point 
, g is defined on an
interval I which contains the range of f and g is
differentiable at the point 
. If 
for 
, show that
h is differentiable at x.
5.
Show that 
on 
if and only if for every 
there exists a partition p
such that 
.
6. State and prove Wierstrass test for uniform convergence.
7. Show that a linear operator A on a finite dimensional vector space X is one-to-one if and only if the range of A is all of X.
8.
Show that a linear operator A on 
is
invertible if and only if 
.
PART B (5 10 = 50 marks)
Answer any FIVE questions.
9.
Show that every nonempty perfect set in 
is
uncountable.
10. Let
be a series of real numbers
which converges, but not absolutely. Let 
.
Show that there exists a rearrangement 
with
partial sums 
such that
; 
.
11. Let
E be a non compact set in 
. Show that
(a) There exists a continuous function on E which is not bounded.
(b) There is a continuous and bounded function on E which has no maximum.
If E is bounded show that there exists a continuous function on E which is not uniformly continuous.
12. State and prove LHospitals rule.
13. Let
f be a bounded function on and 
increases monotonically such
that 
on . Show that if and only if 
and 
.
14. Let
be a sequence of functions
differentiable on such that 
converges for some point 
. If 
converges uniformly on show that converges uniformly on to a function
f and 
.
15. State and prove Stone Weirstrass theorem.
16. State and prove Implicit function theorem.
| Earning: Approval pending. |
