M.Phil. DEGREE EXAMINATION, APRIL 2016
Mathematics
FUNCTIONAL ANALYSIS
(2011 onwards)
Time 3 Hours Maximum 75 Marks
Answer all questions.
15 75)
1. Prove that every topological vector space is a
Hamsdorft space.
Justify that infinite-dimensional topological
vector space is locally compact.
Or
Define: Seminorm on a vector space X. Show
that in every locally convex space exists a
separating family of continuous seminorms.
Show that is a Fve'chet space with the
Heine-Borel property.
2. State and prove category theorem.
State and prove the uniform boundedness
principle.
Or
State and prove the closed graph theorem.
Prove that a bilinear mapping is continuous if
it is continuous at the origin.
Sub. Code
571102
RW-11075
2
ws18
3. State and prove the separation theorem.
Define: Weak topology of a topological vector
space. Also prove that the weak closive
Ew of
a convex subset E of a locally convex space is
equal to its orginal closure E

Or
State and prove Banach-Aloglu theorem.
Show that the convex hull of a compact set in
IRn is compact.
4. state and prove any one of the closed range
theorem.
Define: Annillators and where M is
a subspace of a Banach space X and N is a
subspace of the normed dual of X. Also
describe the duality between these two types
of annihilators.
Or
Define: Compact operator. Illustrate.
State and prove any two properties of the
spectrum of compact T .
5. State Pick-Nevanlinna interpolation problem.
Show that if


2
0 1 z0 z1` 1 and



3
0,2 w0 w1 this problem admits no
solution.
Show that if Pick-Nevanilnna problem has
solutions then it contains a finite Blaschke
product.
Or
Define: Hear measure on compact Groups.
State and prove Kakutani's fixed point
theorem.
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