Vinayaka Missions University 2008 M.Phil Mathematics REAL AND COMPLEX ANALYSIS : - Question Paper

COURSE CODE - 6030502
M.PHIL DEGREE exams – JUNE 2008
M.PHIL (MATHEMATICS)
PAPER – III
REAL AND COMPLEX ANALYSIS
Time: three Hours Max. Marks: 100
PART – A
ans All the ques. 10 X 3=30
1. describe Borel Measures.
2. describe p L spaces.
3. elaborate the properties of Borel Measures?
4. describe Orthonormal sets.
5. State Riesz representation theorem in Complex Measures.
6. State the Fubini’s Theorem.
7. Prove for every f L? , 1, ? ? =
0
0 h
h
t f f
?
l - =
8. elaborate the singularities.
9. describe completion of product measures.
10. State Plancherel Theorem
PART - B
ans any 5 ques. five X six =30
11. If X ' is seperable, then X is separable.
12. If { } n S is a sequence of Measurable sets, then
1
n
n
S S
8
=
=U is
measurable.
13. State and prove Bessel’s Theorem for Founier series.
14. State and prove Nikodym Theorem.
15. If f : [0,a] ?R be bounded and integrable and monotonic on [0,h]
when ( ) ( )
0 0
sin sin
0 , lim 0 one .
a
n
nx x
h a then f x dx f dx
x x
8
< < ? = + ?
16. If { } one two , ,.... n x x x is an orthonormal set, then for any
( )
2
2
1
, ,
n
i
i
x X x x x
=
? S =
17. State and prove Laurent’s expansion
18. State and prove plancherel Theorem
PART - C
ans any 2 ques. two X 20= 40
19. State and prove Ha hn – Banach Theorem
20. State and prove open mapping theorem
21. State and prove Fubini’s Theorem

Earning:   Approval pending.