TU(v) = T(U(v)) = v

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

U=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