= Solution
{c}
Formal self-adjointness gives symmetry of $B$, and strict positivity gives positive definiteness, so $B$ is an <inner product>. Bounded coefficients give $B[u,u]\leq C\|u\|_{H^1}^2$. To prove the lower bound, uniform ellipticity gives
$$
B[u,u]\geq\theta\|Du\|_2^2-\|c_-\|_\infty\|u\|_2^2.
$$
If no lower bound by the $H^1$ norm existed, choose $u_j\in H_0^1$ with $\|u_j\|_{H^1}=1$ and $B[u_j,u_j]\to0$. By <weak convergence in a Hilbert space> and the <Rellich-Kondrachov compactness theorem>, a subsequence has $u_j\rightharpoonup u$ in $H_0^1$ and $u_j\to u$ in $L^2$. The principal energy has <weak lower semicontinuity>: it is the squared $L^2$ norm of the bounded multiplication operator $a^{1/2}Du$. The potential term converges because $c$ is bounded. Hence
$$
B[u,u]\leq\liminf_jB[u_j,u_j]=0.
$$
Strict positivity forces $u=0$. The preceding ellipticity estimate now gives $\|Du_j\|_2\to0$, while $\|u_j\|_2\to0$, contradicting $\|u_j\|_{H^1}=1$. Thus <strict positivity implies coercivity for an elliptic Dirichlet form>, and
$$
\boxed{\alpha\|u\|_{H^1}^2\leq B[u,u]\leq C\|u\|_{H^1}^2.}
$$
The new norm is equivalent to the original norm, so the space remains complete.
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