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:
Let X be a normed space and Y be proper subspace of X. Show that the interior Y0 of Y is empty.
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.
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'.
Give an example of a positive operator on
Given an example of an Hilbert space H and an operator A on H such that is empty. Justify your choice of example.
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).