Determine whether each of the following transforms on L 2 is bounded or not and whether its
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Question:
Determine whether each of the following transforms on L2 is bounded or not and whether its range is closed in L2 or not. For the unbounded transforms, construct a sequence ?of unit-norm functions for which ||Txn||2 is unbounded from above. For the open-range transforms, construct a sequence ?of unit-norm functions for which ||Txn||2 is unbounded from below.
a. T{x}(t) = x(?t) for some ? ? 0.
b. T{x}(t) = x(t3).
c. T{x}(t) = x(t1/3).
d. T{x}(t) = x?(t).
e.
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