Let and
Find
for P = 4 and
.
To find the \( L^P \) norms for the function \( x(t) = t^2 \) on the interval \([0,1]\) for \( P = 4 \) and \( P ________ _________ _____ __________ ___ ____ ________ ___ _________ _________ ____.
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Let {An} be a sequence of unitary operators in BL(H). Prove that if A then A is unitary.
Consider the space 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 Df = f'. 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.
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.
State giving reasons, if the following statement are true or false.
a) A closed map on a normed space need not be an open map.
b) c00 is a closed subspace of
c) The dual of a finite dimensional space is finite dimensional.
d) If T1 and T2 are positive operators on a Hilbert space H, then T1 + T2 is a positive operator on H.
e) On a normed space X, the norm function is a linear map.
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 .
Let X be an inner product space and Prove that
if and only if
Which of the following maps are open? Give reasons for your answer.
i) given by T(x,y,z) = (x,z)
ii) given by T(x,y,z) = (x,y,0)