Prove the following result:
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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.]
) C0 is a Banach space.
Let X be a normed space and Y be proper subspace of X. Show that the interior Y0 of Y is empty.
Give an example of a compact linear map on l2.
Prove the following result:
Let be a sequence of unitary operators in BL(H). Prove that if
, then A is unitary.
Consider the space c00. For , define
. Show that f is a linear functional which is not continuous w.r.t the norm
.