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.
In the proof of the closed graph theorem, we consider a linear operator (F: X \to Y) between Banach spaces (X) and (Y), and assume that the graph of (F), (G(F) = {(x, F(x)) : x \in X}), is closed in (X \times Y). The goal is to show that (F) is continuous. i) Use of (X) being a Banach space: We define a sequence ((x_n)) in (X) such that (x_n \to x) in (X) and (F(x_n) \to _______ ________ ______ _____ ___ ________ _________ __________.
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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.
) Which of the following maps are open? Give reasons for your answer.
i) given by
.
ii) given by
.
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?
Give an example of a positive operator on
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).
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'.
) Let with
. Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.