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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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.
Prove the following result:
Prove the following result:
Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set of a non-zero real numbers with
and an orthonormal set
in H such that
Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?
Given an example of an Hilbert space H and an operator A on H such that is empty. Justify your choice of example.
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
Let X be a normed space and Y be proper subspace of X. Show that the interior Y0 of Y is empty.
Give an example of a positive operator on