MA/MSCMT 02
December Examination 2016
M.A. M.Sc. (Previous) Mathematics Examination
Real Analysis and Topology
Paper MA/MSCMT 02
Time 3 Hours Max. Marks 80
Note: The question paper is divided into three sections B and C. Use of non-programmable scientific calculator is allowed in this paper.
Section A 8 × 2 16
(Very Short Answer Questions)
Note: Section contain Very Short Answer Type Questions. Examinees have to attempt all questions. Each question is of 02 marks and maximum word limit may be thirty words.
Define s ring.
Define measurable set.
State Weierstrass's theorem.
State Fatou's lemma for measurable function.
Define Hilbert space.
231
MA/MSCMT 02 1900 3 (P.T.O.)
MA/MSCMT 02 1900 3 (Contd.)
231
State Parseval's identity.
Define normal space.
(viii) Define directed set.
Section B 4 × 8 32
(Short Answer Questions)
Note: Section contain 08 Short Answer Type Questions. Examinees
have to delimit each answer in maximum 200 words.
Let be an algebra of sets of a set X and be a sequence of
sets in Then show that there exists a sequence of sets in
such that Bi
w Bj if i j and W i 1
3
Bi W i 1
3
Ai.
Let E be a measurable set. Then for any real number prove
that the translation E x is also measurable. Further, prove that.
Prove that every bounded measurable function f defined on a
measurable set E is L-integrable over E.
Prove that the space L2 of square summable functions is linear
space.
Is A B A B Give reason in support of your answer.
Prove that a one one-one onto continuous map f "
is a homeomorphism if f is either open or closed.
Show that T3 space is a topology on X3.
Prove that the product space X × Y is connected if and only if X
and Y are connected.
MA/MSCMT 02 1900 3
231
Section C 2 × 16 32
(Long Answer Questions)
Note: Section contain 04 Long Answer Type Questions. Examinees
will have to answer any two questions Each questions is of
16 marks. Examinees have to delimit each answer in maximum
500 words.
10) Prove that continuous image of a connected space is
connected.
State and prove Alexander subset lemma for compact
topological space.
11) Prove that every T3 space is a T2 space.
State and prove Minkowski's inequality.
12) Prove that the necessary and sufficient condition for a bounded
function f defined of interval to be L-integrable over
is that for given R there exists a measurable partition
P of such that U R.
13) State and prove Riesz-Fisher theorem.