MA/ MSCMT-06
December Examination 2017
M.A./ M.Sc. (Final) Mathematics Examination
Analysis and Advanced Calculus
Paper MA/ MSCMT-06
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 Type Questions)
Note: Section contain 8 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 Bounded Linear Transformation for Normed Vector Space.
Define Second Dual Space.
State Pythagorean Theorem.
Define Eigen Values for a Hilbert Space.
Define C1 maps.
Define Locally Lipschitz Function.
154
MA/ MSCMT-06 1400 3 (P.T.O.)
154
MA/ MSCMT-06 1400 3 (Contd.)
Define Self Adjoint Operator.
(viii) State Polarisation identity in a Hilbert Space.
Section B 4 × 8 32
(Short Answer Type Questions)
Note: Section contain 08 short Answer Type Questions. Examinees
will have to answer any four question. Each question is of
08 marks. Examinees have to delimit each answer in maximum
200 words.
State and prove Minkowaski's Inequality.
Let N and N′ be normed linear space and D be a subspace of N. Then
prove that a linear transformation T D → N′ is closed iff its graph
TG is closed.
Let M be a closed linear subspace of a Hilbert space H. Let x be a
vector not in M and d M). Then prove that there exist a unique
vector y0 in M s.t. x y
0 d.
Show that the set of unitary operators on a Hilbert space forms a
multiplicative group.
State and prove Mean value theorem for Banach space.
Let f be a regulated function on a compact interval of R into
R such that a b and for all t in f 0. Then prove that
f(t)dt 0 a
b
. Further prove that if f be a continuous function at a
point c of b and f then f(t)dt 0 a
b 2 .
154
MA/ MSCMT-06 1400 3
Let I be an open interval of R and W be an open subset of a Banach
space X over K. Let x0) be a point of I × W and let g be a
continuous map of I × W into X. Then prove that a continuous map
I → W is an integral solution for g at x0) iff for each t ∈
x0
t
t
0

Show that every compact subset of a normed linear space is bounded
but its converse need not be true.
Section C 2 × 16 32
(Long Answer Type Questions)
Note: Section contain 04 Long Answer Type Questions. Examinees
will have to answer any two questions. Each question is of
16 marks. Examinees have to delimit each answer in maximum
500 words.
10) State and prove Natural embedding theorem for normed linear space.
11) State and prove Risez representation theorem Hilbert space.
12) State and prove Spectral theorem for finite dimensional Hilbert
space.
13) State and prove Implicit function theorem on differentiable functions
over Banach space.