7. a) Let x = c, with ||- Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.

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Please answer (a) only. This is a FUNCTIONAL ANALYSIS question. I will upvote surely.

7. a)
Let X = c 0
with . Give an example of a Cauchy sequence in X that do not converge
in X. Justify your choice of example.
b)
Give one example of each of the following. Also justify your choice of example.
i)
A self-adjoint operator on 2.
ii)
A normal operator on a Hilbert space which is not unitary.
c)
Let X be a normed space and Y be proper subspace of X. Show that the interior Y ° of Y
is empty.
Transcribed Image Text:7. a) Let X = c 0 with . Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example. b) Give one example of each of the following. Also justify your choice of example. i) A self-adjoint operator on 2. ii) A normal operator on a Hilbert space which is not unitary. c) Let X be a normed space and Y be proper subspace of X. Show that the interior Y ° of Y is empty.
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