= Solution
The functional
$$
\ell(\varphi)=-\int_\Omega f\varphi
$$
is bounded on $H_0^1(\Omega)$ by the <Cauchy-Schwarz inequality> and the Poincare inequality:
$$
|\ell(\varphi)|\leq C\|f\|_{L^2}\|\nabla\varphi\|_{L^2}.
$$
The <Riesz representation theorem> therefore supplies a unique $u\in H_0^1(\Omega)$ satisfying
$$
(u,\varphi)_{H_0^1}=\ell(\varphi)
$$
for every test function. This is exactly the weak identity from part (a). Uniqueness also follows by testing the homogeneous difference with itself.
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