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)) and (y = (y_n)) in (\ell^2), we have ____ ________ _______ ________ _____.
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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.]
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'.
Prove the following result:
) Which of the following maps are open? Give reasons for your answer.
i) given by
.
ii) given by
.
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.
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.
Consider the space c00. For , define
. Show that f is a linear functional which is not continuous w.r.t the norm
.