State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
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Prove the following result:
Let X be an inner product space and . Prove that
if and only if
.
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.
) Which of the following maps are open? Give reasons for your answer.
i) given by
.
ii) given by
.
Give an example of a positive operator on
et and F be the set of all
in H such that
. Find
. Verify that every
can be expressed as
where
and
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
b) C0 is a Banach space.
c) If A is the right shift operator on l2, then the eigen spectrum is non-empty.
d) If a normed linear space is reflexive, then so is its dual space.
e) If a normed linear space X is finite dimensional, then so is X'.