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.
i) Consider the operator (T: \ell^2 \to \ell^2) defined by (T(x_1, x_2, x_3, \dots) = (x_1, 2x_2, 3x_3, \dots)). This operator is diagonal with real entries (1, 2, 3, \dots) on the diagonal. For any (x = (x_n)) ________ ______ ___ __________ _________ ______.
_______ ____ ___ ___ ___ ____ ________ _________ ______ __________ __________.
____ ____ _____ ________ __________ ___ ____ ___ ________ __________.
_____ __________ _____ ________ ______ _______ __________ __________ ______ ___.
__________ _____ _____ _______ ____ ____ _______ ______ _______ _________ _____ ____.
________ _____ _________ __________ _______ _____ _________ ______ _____.
_____ ___ _________ ___ _________ _______ _____ ______ ___ ________ _____.
________ _________ __________ ______ __________ ______ ____ ______ _________ ___.
____ _______ __________ ________ _______ ____ _____ _____ _____.
_______ ___ ____ ___ _________.
_______ ___ ______ _______ ________ ____ ________ _________ ____ _________ _______.
________ ______ ____ ____ ____ ___ ________ _______.
______ ______ _____ _____ _________ _______.
______ __________ ___ ________ ________ ________ _________.
__________ ___ ___ ____ _______ _________ ____ ______ ________ __________.
________ _____ _________ _______ __________.
________ _______ _________ _________ ______ _______ ___.
_________ ___ ___ ___ _______ ________ ____ ____ ___ ___ __________.
__________ __________ ________ __________ __________ ___.
_______ ____ ____ _________ _______ _____ ____ ____.
________ _________ _______ ________ ____ ________ _____ _____.
________ ______ ___ _______ ________ ______ _________ ____ ________ ________ ___.
____.
Get Full Answer on WhatsApp
Let X be a Banach space and Y be a closed subspace of X. Let be canonical quotient map. Show that
is open.
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-schmidt operator a compact operator? Justify your answer.
Give an example of a positive operator on
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.
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.
When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]