Give an example of a positive operator on .
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Let A be a normal operator on a Hilbert space X. Show that σ(A) ⊂ σa(A) where σa (A) denotes the approximate eigen spectrum of A and σ(A) denotes the spectrum of A.
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 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 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.
Let X be a normed space and Y be proper subspace of X. Show that the interior Yº of Y is empty.
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.]
Give an example of a compact linear map on l2.
Given an example of an Hilbert space H and an operator A on Η such that σe(A)is empty. Justify your choice of example.