Before Theorem 6 in Chapter 7.4 in Evans' PDE book Evans claims that there exists $\beta > 0$ such that $$ \beta\|u\|_{H^1(\Omega)}^2 \leq B[u,u]\,, \quad \forall u \in H_0^1(\Omega)\,. $$ From how it's stated I get the vibe that this should be trivial but for some reason I can't figure out the reasoning.
Here $H_0^1(\Omega)$ is the closure of smooth compactly supported functions in $H^1(\Omega)$ and $\Omega \subset \mathbb{R}^n$ is open, bounded, and has smooth boundary.
$B$ is the bilinear form associated with a hyperbolic operator $\partial_t^2 + L$ where $$ Lu = -\sum_{i,j=1}^n (a^{ij}(x,t) u_{x_i})_{x_j} + c(x,t) u $$ ($c \geq 0$ and $a^{ij} = a^{ji}$, I guess $a^{ij}$ and $c$ are continuously differentiable too, but he's not very clear with his assumptions) and thus $$ B[u,v] = \int_\Omega \left( \sum_{i,j=1}^n a^{ij}(x,t) u_{x_i} v_{x_j} + c(x,t) uv \right) \, dx\,. $$ Hyperbolcity of $\partial_t^2 + L$ means that there exists $\theta > 0$ such that $$ \sum_{i,j=1}^n a^{ij} \xi_i \xi_j \geq \theta |\xi|^2 $$ for all $\xi = (\xi_1, ..., \xi_n) \in \mathbb{R}^n$.
I think the assumption $u \in H_0^1(\Omega)$ hints towards integration by parts or the Poincaré inequality ($\|u\|_{L^2} \leq C \|Du\|_{L^2}$).
This seems to work:
The hyperbolicity implies $$ \theta \int_\Omega |Du|^2 \, dx \leq \int \sum_{i,j} a^{ij} u_i u_j \, dx = B[u,u] - \int_\Omega c u^2 \, dx \leq B[u,u]\,, $$ since $c \geq 0$.
For $u \in H_0^1(\Omega)$ the Poincaré inequality implies that there exists $C \geq 0$ such that $$ \|u\|_{H^1(\Omega)}^2 = \|u\|_{L^2(\Omega)}^2 + \|Du\|_{L^2(\Omega)}^2 \leq (C+1)\|Du\|_{L^2(\Omega)}^2\,. $$ Combining these we get $$ \|u\|_{H^1(\Omega)}^2 \leq \frac{C+1}{\theta} \theta \|Du\|_{L^2(\Omega)}^2 \leq \frac{C+1}{\theta} B[u,u]\,. $$ So we may choose $\beta = \frac{\theta}{C+1}$.