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'.
a) False. Not every function equation defines a norm. For a function to be a norm, it must satisfy positivity, scalability, and the triangle inequality. ____ _________ _________ _____ ____ _____ ______ ___ ________.
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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 X be a Banach space, Y be a normed linear space and be a subset of B(X, Y). If
is not uniformly bounded, then there exists a dense subset D of X such that for every
is not bounded in Y.
Let X, Y be normed spaces and suppose BL(X, Y) and CL(X, Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X, Y) is linear subspace of BL(X, Y). Also, Show that if Y is a Banach space, then CL(X, Y) is a closed subspace of BL(X, 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.
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:
et and F be the set of all
in H such that
. Find
. Verify that every
can be expressed as
where
and