MA/MSCMT-06
December 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. 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.
Write uniqueness theorem of differential equations for Banach space.
Define regulated function for Banach space.
Define orthogonal projection for Hilbert spaces.
Define complete orthonormal set for Hilbert space.
Define inner product space.
154
MA/MSCMT-06 1000 3 (P.T.O.)
MA/MSCMT-06 1000 3 (Contd.)
154
Define first dual space.
Define multilinear mapping.
(viii) Explain weak convergence.
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.
State and prove Holder's inequality for normed linear space.
State and prove Reisz lemma for normed linear space.
Prove that a closed convert subset K of Hilbert space H contains
a unique vector of smallest norm.
Prove that if M and N are closed linear subspaces of Hilbert
space H such that M N then then the linear subspaces M N
is closed.
Prove that every Hilbert space is reflexive.
Prove that if T is normal operator on a Hilbert space H then
eigen spaces of T are pair wise orthogonal.
If X and Y are two banach spaces over the same field K of
scalers and V is an open subset of X. Let f V "Y be continuous
functions. Let v be any two distinct points of V such that
X v and if f is differentiable in then prove that.
f f v u sup " Df x
State and prove Global uniqueness theorem.
MA/MSCMT-06 1000 3
154
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) State and prove closed graph theorem for normed linear space.
11) State and prove Hahn-Banach theorem for normed linear spaces.
12) State and prove implicit function theorem on differentiate
functions over Banach sapces.
13) If f be a function on a compact interval of R into a Banach
space X over K.