Gauhati University 2007 Previous : thetics--II - Question Paper
Monday, 21 January 2013 08:40Web
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(b) l1
(c) l3
(d) l4
11. Let r1 be the radius of the closure of the open ball B(x0, r). Then
(a) r1> r
(b) r1< r
(c) r1= r
(d) None of (a), (b) and (c)
12. In an inner product space if A B, then
(d) None of (a), (b) and (c)
13. Orthonormal set derived from x1= (3, 0, 4), x2 = (-1, 0, 7) and x3 = (2, 9, 1)
is
T T I x if one two (b) Ì
T T I x if two one (c) Ì
( 4,0,3), (0,1,0)
5
1
(4,0,3),
5
1
(b) -
(2,9,1)
85
1
( 4,0,3),
5
1
(c) (1,0,0), -
(2,9,1)
85
1
(1,1,0),
2
1
(d) (3,0,4),
(a) (T ) one (T 1)* * - - =
b T T T T*
2
*
1
*
1 2( ) ( ) =
(c) (aT)* =aT *
(d) Sx, x = x,Tx S* = T *
2 two 2 (a) x + y > x + y
2 two 2 (b) x + y < x + y
2 two 2 (c) x + y = x + y
n n u, v = u v + u v + ..... + u v one 1 two 2
Ã
14. If T* be adjoint operator of T, then
15. In an inner product space if xy, then
(d) None of (a), (b) and (c)
16. In inner product space (Cn, .) with
if u = (i, i, i, …i), then u is equal to
(a) n (b) n (c) n2 (d) none
PART-B (Subjective-type Question)
(Marks: 48)
ans any 3 parts of every ques. (17-20): 12×4 = 48
17. (a) Consider the topology T= {, {a}, {c, d}, {a, c, d},{b, c, d, e}, X} on the set X =
{a, b, c, d, e}. obtain the relative topology of T on A = {a, d, e}.
(b) Show that the class = { [ a , b )| a, b rational, a
¥
=
¥
=
= =
1
2 2
1
, ,
j
j j
j
j x x e e and x x e
T *x -m x = Tx -mx
R.
(c) Prove that a mapping f from a topological space X to a topological space Y is
continuous if and only if for every subset AX.
(d) Prove that every metric space is 1st countable.
(e) Prove that every regular T1 space is a T2 space.
18. (a) Show that a metric space is compact if and only if it is sequentially compact.
(b) Show that every compact space has the Bolzano - Weierstrass property.
(c) Prove that a set A is connected if and only if A is not union of 2 non-empty
separated sets.
(d) Let A be a connected subset of topological space X. If
then show that B is also connected.
(e) Prove that every projection on a product space
is both open and continuous
19. (a) In a normed linear space if A is compact and B is closed, then prove
that A+B is closed
(b) Prove that T: l
l
is a bounded linear mapping.
(c) Prove that 2 norms on a linear space X are equivalent if
and only if there exist 2 numbers a, b>0 such that
(d) Show that dual space of c0 is l1.
(e) State and prove ‘the closed graph theorem’.
20. (a) Show that in an inner product space the
parallelogram legal regulations holds.
b) Prove that a non-empty closed convex set C in a Hilbert space has a
unique point of minimal norm.
c) If M is a closed subspace of a Hilbert space H, then prove that X=M M.
d) Let X be a Hilbert space and (ei) a complete orthonormal sequence. Then
for an x ?X, prove that
(e) If T is normal and µ is scalar, then show that T-µI is normal and
for all x ?X.
A Ì B Ì A
i i P : X ® X i
i
X = P X
(X, . )
,...)
3
,
2
,
1
( , , ,...) ( one two 3
1 two 3
x x x
x x x x Tx = ® =
1 two . and .
. one two one a x £ x £ b x for all xÎ X
(X , ),
| Earning: Approval pending. |
