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PG–0213
MMS–25
M.Sc. DEGREE EXAMINATION
DECEMBER 2010.
Second Year
(AY – 2006-07 batch onwards)
Mathematics
TOPOLOGY AND FUNCTIONAL ANALYSIS
Time : 3 hours
Maximum marks : 75
SECTION A — (5  5 = 25 marks)
Answer any FIVE questions.
1.
Prove that intersection of two topologies on a set X
is a topology on X.
2.
Let Y be a subspace of X. If A is closed in Y and Y
is closed in X, prove that A is closed in X.
3.
Let A be a connected subset of X. If A  B  A
prove that B is also connected.
4.
Prove that a subspace of a Hausdorff space is
Hausdorff.
5.
Prove that the image of compact space under a
continuous map is compact.
6.
State and prove closed graph theorem.
7.
State and prove Schwarz inequality.
8.
If T is an operator on H for which (Tx , x )  0 for
all x , prove that T  0.
SECTION B — (5  10 = 50 marks)
Answer any FIVE questions.
9.
Let ( X , T1 ) and (Y , T2 ) be two topological spaces.
Let   U  V , where U is T1 open in X and V is
T2 open in Y. Prove that  is a basis for a
topology on X  Y.
10.
Prove that every finite set in a Hausdorff space is
closed.
11.
Prove that a space X is connected if and only if the
only subsets of X that are both open and closed in
X are the empty set and X itself.
12.
Suppose that X has a countable basis. Prove that
there exists a countable dense subset of X.
13.
State and prove Urysohn lemma.
14.
Describe the Hahn–Banach theorem.
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15.
Prove that a closed convex subset C of a Hilbert
space H contains a unique vector of smallest norm.
16.
If T is an operator on H, prove that T is normal if
and only if its real and imaginary parts commute.
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