Tamil Nadu Open University (TNOU) 2009-2nd Year M.Sc Mathematics Tamilnadu open university Maths Functional analysis - Question Paper
M.Sc. DEGREE exam —
JUNE, 2009.
(AY 2005–06 and CY 2006 batches only)
Second Year
Mathematics
FUNCTIONAL ANALYSIS
Time : three hours Maximum marks : 75
part A — (5 x five = 25 marks)
ans any 5 ques..
1.If M is a closed linear subspace of a normed linear spare N and one and T is the natural mapping of N onto N/M described by , show that T is a continuous linear transformation for which .
2.State and prove Uniform boundedness theorem.
3.Let M be a closed liner subspace of a Hilbert
space H, let and let d be the distance from
x to Show that there is a unique vector such that .
4.If A is a positive operator on a Hilbert space H show that is non singular.
5.Let B be a basis for H and T an operator whose matrix relative to B is . Show that T is nonsingular if and only if is non singular.
6.Let R be the radical of a Banach algebra A. Show that if is regular for and , is also regular.
7.Let A be a commutative Banach Algebra show
that the subsequent conditions are equivalent
(a)for
(b)for
(c)for .
8.Let X and Y be nls and be linear. Show that F is compact if and only if for every bounded sequence in X, contains a subsequence which converges in Y.
part B — (5 × 10 = 50 marks)
ans any 5 ques..
9.Let M be a linear subspace of a normed linear space N and f be a functional described on M. If and if show that f can be extended to a functional described on such that .
10.If T is an operator on a normed linear space N, show that its conjucate is an operator on and the mapping is an isometric isomorphism of into which reverses products.
11.If is an orthonormal set in a Hilbert space H, and if show that for every j.
12.(a)If T is an operator on Hilbert space H, show that T is normal if and only if its real and imaginary parts commute.
(b)Show that an operator T is unitary if and only if it is an isometric isomorphism of H onto itself.
13.Let T be a normal operator on a Hilbert space H with eigen values and eigen spaces . Show that ’s are pairwise orthogonal, every decreases T and ’s span H.
14.For every x in a Banach Algebra A, show that the spectral radius .
15.State and prove Gelfand Neumark Theorem.
16.(a)Define compact linear map and show that collection of all compact linear maps is a subspace of .
(b)Show that the spectrum of a compact map A is countable.
—–––––––––
M.Sc. DEGREE EXAMINATION
JUNE, 2009.
(AY 200506 and CY 2006 batches only)
Second Year
Mathematics
FUNCTIONAL ANALYSIS
Time : 3 hours Maximum marks : 75
SECTION A (5 5 = 25 marks)
Answer any FIVE questions.
1.
If M is a closed linear subspace of a normed linear spare N
and 1 and T is the natural mapping of N onto N/M
defined by 
, show that T is a
continuous linear transformation for which 
.
2. State and prove Uniform boundedness theorem.
3.
Let M be a closed liner subspace of a Hilbert
space H, let 
and let d be
the distance from
x to 
Show that there is a unique
vector 
such that 
.
4.
If A is a positive operator on a Hilbert space H show that
is non singular.
5.
Let B be a basis for H and T an operator whose
matrix relative to B is 
. Show that
T is nonsingular if and only if is
non singular.
6.
Let R be the radical of a Banach algebra A. Show that if 
is regular for 
and 
, 
is
also regular.
7.
Let A be a commutative Banach Algebra show
that the following conditions are equivalent
(a) 
for
(b) 
for
(c) 
for .
8.
Let X and Y be nls and 
be
linear. Show that F is compact if and only if for every bounded sequence
in X, 
contains a subsequence which
converges in Y.
SECTION B (5 10 = 50 marks)
Answer any FIVE questions.
9.
Let M be a linear subspace of a normed linear space N and f
be a functional defined on M. If 
and
if 
show that f can be
extended to a functional 
defined on 
such that 
.
10. If
T is an operator on a normed linear space N, show that its
conjucate 
is an operator on 
and the mapping 
is an isometric isomorphism
of 
into 
which reverses products.
11. If
is an orthonormal set in a
Hilbert space H, and if 
show that 
for each j.
12. (a) If T is an operator on Hilbert space H, show that T is normal if and only if its real and imaginary parts commute.
(b) Show that an operator T is unitary if and only if it is an isometric isomorphism of H onto itself.
13. Let
T be a normal operator on a Hilbert space H with eigen values 
and eigen spaces 
. Show that 
s are pairwise orthogonal,
each reduces T and s span H.
14. For
each x in a Banach Algebra A, show that the spectral radius 
.
15. State and prove Gelfand Neumark Theorem.
16. (a) Define
compact linear map and show that collection of all compact linear maps is a
subspace of 
.
(b) Show that the spectrum 
of
a compact map A is countable.
| Earning: Approval pending. |
