Bhavnagar University 2007 M.Sc Mathematics TOPOLOGY AND FUNCTIONAL ANALYSIS - Question Paper

M.Sc. DEGREE EXAMINATION, DECEMBER 2007.
Mathematics
TOPOLOGY AND FUNCTIONAL ANALYSIS
Time : 3 hours Maximum : 100 marks
ans any 5 ques..
All ques. carry equal marks.
1. (a) Show that any closed subset of a topological space X is the disjoint union of its set of isolated points and its set of limit points.
(b) Let S be a class of subsets of a nonempty set X. Show that the class of all unions of finite intersections of sets in S is a topology on X.
2. (a) Show that in a sequentially compact metric space, every open cover has a Lebesgue number.
(b) Prove that a continuous mapping of a compact metric space into a metric space is uniformly continuous.
3. (a) State and prove Jietze extension theorem.
(b) Show that the product of a nonempty class of Hausdorff spaces is a Hausdorff space.
4. (a) Show that the connected subspaces of the real line are precisely the intervals.
(b) Describe the 1 point compactification of a locally compact Hausdorff space X.
5. (a) Let N and be normed linear spaces and T a linear transformation of N into . Prove that T is bounded if and only if it is continuous.
(b) Let M be a linear subspace of a normed linear
space N, a vector not in M and f be a functional described
on M. Prove that f can be extended to a functional on the linear subspace spanned by M and such that .
6. (a) Let B and be Banach spaces and T be a continuous linear transformation of B onto . Prove that the image of every open sphere centred on the origin in B contains an open sphere centred on the origin in .
(b) Deduce open Mapping Theorem.
7. (a) If M is a proper closed linear subspace of a Hilbert space H, show that there exists a nonzero vector in H which is orthogonal to M.
(b) If is an orthonormal set in a Hilbert space H, prove that for every vector x in H.
8. (a) Let H be a Hilbert space and f be an arbitrary functional in . Prove that there exists a unique vector y in H such that for every x in H.
(b) If P is a projection on H with range M and null space N, show that is self adjoint and in this case .

Earning:   Approval pending.