A common technique to show existence of weak solutions to the problem $Lu=f$ is to obtain an energy estimate of the form:
$$\| u\|\leqslant\| L^{*}u\| $$
and the define the functional $$k(L^{*}v)=\int fv.$$
Then using the Hanh–Banach theorem together with the Riesz representation theorem one can find a $u$ such that
$$ k(L^{*}v)=\int fv=\int uL^{*}v,$$
which shows that $u$ is a weak solution.
Isn't uniqueness also immediate from the Riesz Representation Theorem?
I usually see that people use again the energy inequality and show that the difference of the two solutions satisfying the same data is zero when dealing with hyperbolic system whereas when the problem is elliptic they use this argument.
Why is that step needed?