Hence range of T is entire V i.e. T is onto V.
As T is onto AND V is finite dimensional => T is invertible.
TU = I
(T^-1)TU = T^-1
As for the example in infinite case - can't think. But I guess there was something like
T(q(x)) = q'(x)
U(q(x) = Integral from 1 to x q(x)
Please check it though
Here V is F[x] - all polynomials in x