Gauhati University Topology and Functional Analysis 2007 Question Paper

University :  Gauhati University
Degree : B.Sc Mathematics
Question Paper Subject: Topology and Functional Analysis
Question Paper Year : 2007
Question Paper Details : Previous Years Question Papers Types Version

(Topology and Functional Analysis)

The figures in the margin indicate full marks for the questions

PART-A (Objective-type Questions)

(Marks: 32)

Each question (1-16) carries four codes (a), (b), (c) and (d), out of which one is for

correct answer. Choose the correct code: 2×16=32
1. Let X be a set and Tic be the collection of all subsets U of X such that X-U is
either countable or is all of X. Then the topology Tic on X is called
a) Cofinite
b) Cocountable
c) Discrete
d) Indiscrete
2. (R, u) is the usual topological space on R. Then the closure of the set of natural
numbers (N) is
(a) R
(b) N
(c) N U{0}
(d) None
3. Let A = (a, b), B = [a, b) and C = (a, b]. Then
4. Let (R, u) be usual topological space and T be relative topology on [0, 1].
Then
(a) (½, 1] is u-open but not T-open
(b) (½, 1] is both u-open and T-open
(c) (½, 1] is neither u-open and nor T-open
(d) (½, 1] is not u-open but T-open
5. Let X = {1, 2, 3, 4, 5} and T = {
, {1}, {2, 3} {1, 2, 3}, X} is a topology on
X. Let B1 = {{1, 2}, X}, B2 = {{2, 3}, X}, B3 = {{1, 2, 3}}. Then
(a) B1 is a local base at 1
(b) B2 is a local base at 3
(c) B2 is a local base at 2
(d) B2 is not a local base at 2
T T I x a if 1 2 ( ) Ì
a A B^ ^ ( ) =
b A B^ ^ ( ) Í
c B A^ ^ ( ) Í
( 4,0,3), (0,1,0)
5
1
(3,0,4),
5
1
(a) -
6. Let Ix: (X, T1) (X, T2) be identity map from topological space (X, T1) into
(X, T2). Then
is an open map
is a continuous map
is open
(e) None of (a), (b) and (c)
7. If R is equipped with usual topology, then
(a) (0,1) is a compact subset of R
(b) (0,1) U[1,2) is a connected set
(c) {0, 1} is a compact subset of R
(d) [0, 1) is a compact and connected set
8. (a) A Hausdorff space is metrizable
(b) A metric space is a Hausdorff space
(c) Hausdorff property is not hereditary
(d) Hausdorff is not a topological property
9. Continuity of linear function is characterised by
(a) Hahn-Banach theorem
(b) The open mapping theorem
(c) The closed graph theorem
(d) The principle of uniform bounded ness
10. The law of parallelogram is nor satisfied by
(a) l2
(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 1 2 (b) Ì
T T I x if 2 1 (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 ) 1 (T 1)* * - - =
b T T T T*
2
*
1
*
1 2( ) ( ) =
(c) (aT)* =aT *
(d) Sx, x = x,Tx
S* = T *
2 2 2 (a) x + y > x + y
2 2 2 (b) x + y < x + y
2 2 2 (c) x + y = x + y
n n u, v = u v + u v + ..... + u v 1 1 2 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)
Answer any three parts of each question (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}. Find the relative topology of T on A = {a, d, e}.
(b) Show that the class = { [ a , b )| a, b rational, af (A) Í f (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 A
X.
(d) Prove that every metric space is first 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 two 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 two norms on a linear space X are equivalent if
and only if there exist two 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 law 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
( , , ,...) ( 1 2 3
1 2 3
x x x
x x x x Tx = ® =
1 2 . and .
. 1 2 1 a x £ x £ b x for all xÎ X
(X , ),

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