State giving reasons, if the following statement are true or false.
a) A closed map on a normed space need not be an open map.
b) c00 is a closed subspace of
c) The dual of a finite dimensional space is finite dimensional.
d) If T1 and T2 are positive operators on a Hilbert space H, then T1 + T2 is a positive operator on H.
e) On a normed space X, the norm function is a linear map.
a) True. A map between normed spaces is said to be closed if it maps closed sets to closed sets. However, it need not map open sets to open sets. For example, consider the identity map from the real numbers with the standard topology to itself. It is closed but not open.
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Define the spectral radius of a bounded linear operator .Find the spectral radius of A in
,where A is given by the matrix
with respect to the standard basis of .
Let and F be the set of all
in
such that
. Find
Verify that every
can be expressed as
where
and
.
Let and
Find
for P = 4 and
.
Let X be a Banach space and Y be a closed subspace of .X Let be canonical quotient map. Show that
is open.
Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.
i) X is a Banach space.
ii) Y is a Banach space.
iii) F is a closed map.
iv) Which property of continuity is being established to conclude that F is continuous.
Let X be a Banach space, Y be a normed linear space and f be a subset of B(X,Y).If f is not uniformly bounded, then there exists a dense subset D of X such that for every is not bounded in Y.
Give one example of each of the following. Also justify your choice of example.
i) A self-adjoint operator on .
ii) A normal operator on a Hilbert space which is not unitary.