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 ||Tx_n||² is unbounded from above. For the open-range transforms, construct a sequence ?of unit-norm functions for which ||Tx_n||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(t^1/3).

d. T{x}(t) = x?(t).

e.

{In(t)}=1 {In(t)}=1 T{r}(t) = f(T)dT