= Strict positivity implies coercivity for an elliptic Dirichlet form
Let $B[u,v]=\int_U a^{ij}\partial_i u\partial_j v+cuv$ on $H_0^1(U)$ for a bounded smooth domain, smooth symmetric uniformly positive coefficients, and bounded real $c$. If $B[u,u]>0$ for every nonzero $u$, then $B[u,u]\geq\alpha\|u\|_{H^1}^2$ for some $\alpha>0$. If this failed, choose $\|u_j\|_{H^1}=1$ with $B[u_j,u_j]\to0$. The <Rellich-Kondrachov compactness theorem> gives a subsequence weakly converging in $H_0^1$ and strongly in $L^2$. The principal energy has <weak lower semicontinuity>, and the potential term converges, so strict positivity forces the limit to be zero. Uniform ellipticity then gives $\|Du_j\|_2\to0$, contradicting normalization. Merely being a <positive semidefinite operator> is insufficient: $-\Delta-\lambda_1$ has a nonzero Dirichlet <eigenfunction> of zero energy.
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