M.Phil. DEGREE EXAMINATION, APRIL 2016
Mathematics
FUNCTIONAL ANALYSIS
(2015 onwards)
Time 3 Hours Maximum 75 Marks
Answer all the questions.
All questions carry equal marks.
15 75)
1. Prove that every topological vector space is a
Hausdorff space.
Prove that every topological vector space has a
balanced convex local base.
Or
Prove that Lp is a locally bounded F -space.
Prove that a topological vector space X is
normable if and only if its origin has a convex
bounded neighbourhood.
2. State and prove Banach-Steinhaus theorem.
Or
State and prove closed graph theorem.
Sub. Code
571102
RW-11073
2
Sp2
3. If X is a locally convex space then prove that
separates points on X .
If f is a continuous linear functional on a
subspace M of a locally convex space X then
there exists such that f on M .
Or
State and prove Krein-Milman theorem.
4. Let M be a closed subspace of a Banach space
X . The Hahn-Banach theorems extends each
to a functional . Define
x . Then prove that is an
isomorphism of onto X .
Prove that every finite dimensional normal
space is reflexive.
Or
If X and Y are Banach spaces and T
Then prove that T is compact if and only if is
compact.
5. Prove that every differentiable operator is
a continuous mapping of into
If in and gi g in then
prove that gi g in
Or
Suppose hj is an approximate identity on Rn
and . Then prove that
j hj lim
limj u hj u