State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
The function is defined as [ |x| = \sum_{j=1}^{n} |a_j|, \quad x = (a_1, a_2, \dots, a_n) \in \mathbb{R}^n. ] To check if this is a norm, we verify the _________ __________ __________ _______ _______ _____ ______ _______.
__________ _______ ___ ________ ___ ___ ________ ______ __________ _________.
___ ____ ______ __________ _________ _________ __________ ________ ______ _____ ______ _______.
___ _________ ______ ______ _____ ___ _____.
____ _________ _________ ______ _____ ______ _________ ________ _______.
____ _________ _______ __________ _____ ___ _____ ________ _________ ___ ___ _______.
_____ _____ _________ ______ ___ _________ __________ ________ _____ _____.
________ _____ _________ ______ _____ _____ ________ __________ ________ ____.
__________ ________ ______ __________ ____ __________ ______.
___ __________ ____ ________ ________ ______ _________ ______ _________ _____.
__________ _________ ____ ____ ____ ___ ___ _______.
________ _______ ________ _____ ________ ___ ____ ________ ___ __________ ______.
_______ ____ ___ ______ ________ _______.
_________.
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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.
) C0 is a Banach space.
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.
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 be an inner product space and . Prove that
if and only if
.
Define the spectral radius of a bounded linear operator . Find the spectral radius of A in
, where A is given by the matrix
with respect to the standard basis of .
Consider the space C1[0,1] of all C1 functions on [0,1] endowed with the uniform norm induced from the space C[0,1], and consider the differential operator defined by
. Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C1[0,1] is not a Banach space? Justify your answer.