MA/MSCMT-06
June Examination 2016
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.
Section A 8 × 2 16
(Very Short Answer Questions)
Note: Section contain 08 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 convergence in normed linear space.
Explain weak convergence.
State open mapping theorem.
Define Hilbert space.
Define orthogonal complement.
Define self-adjoint operators.
State spectral theorem for Hilbert space.
(viii)Write Taylor's formula with integral remainder.
154
MA/MSCMT-06 1100 4 (P.T.O.)
MA/MSCMT-06 1100 4 (Contd.)
154
Section B 4 × 8 32
(Short Answer Questions)
Note: Section contain 08 short answer type questions.
Examinees will have to answer any four questions.
Each question is of 08 marks. Examinees have to delimit
each answer in maximum 200 words.
It T be a linear transformation of a normed linear space N to
normed linear space N′, then prove that inverse of T exists and
is continuous on its domain of definition If and only If there
exists a constant K 0 S.T. K x T 6 xdN
If M is a closed linear subspace of a normed linear space N and
x0 is a vector not in M then prove that there exist a functional F
in conjugate space such that and 0.
Prove that the mapping H " defined by fywhere
fy for every x ∈ H is an additive, one to one onto
isometry but not linear.
If T is an operator on a Hilbert space then prove that
0 6 xdH If and only If T 0.
Let λ be an Eigen value of an operator T on a Hilbert space. If
Mλ is the set consisting of all. Eigen vectors of T corresponding
to Eigen value λ and the zero vector then prove that Mλ is a
non-zero closed linear subspace of H in variant under T.
MA/MSCMT-06 1100 4 (P.T.O.)
154
If X and Y are any two Banach spaces over the same field
K of scalers and V is an open subset of X. Let f V→ Y be
continuous function and u1 v be any two distinct points of V
such that V and f is differentiable in then prove
that f f v u Sup {Df x ∈
If f be a regulated function on a compact interval of R into
a Banach space X over K (field of scalers) and g is continuous
linear map of X into Banach space Y over then prove that
gof is regulated and
State and prove Global uniqueness theorem for locally Lipschitz
functions.
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
question is of 16 marks. Examinees have to delimit each
answer in maximum 500 words. Use of non-programmable
scientific calculator is allowed in this paper.
10) Let P be the real number such that p ∞ show that the space
p
n l of all n-Tuples of scalers with the norm defined by
11) State and prove uniform boundness theorem.
12) If B is a Complex Banach space whose norm obeys parallelogram
law and if an inner product is defined on B by
u
x y x y i x iy i x iy then
prove that B is a Hilbert space.
13) If X and Y be Banach spaces over the same field K of
scalers and V be an open subset of X. Let f V → Y is twice
differentiable at point v ∈ then prove that D2f ∈
is a bilinear symmetric mapping that is for all ∈ XXY
D2f
D2f