Prove the following result:
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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.
Let be given by
. Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.
When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]
Given an example of an Hilbert space H and an operator A on H such that is empty. Justify your choice of example.
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?
Let A be a normal operator on a Hilbert space X. Show that where
denotes the approximate eigen spectrum of A and
denotes the spectrum of A.