Testing a coercive weak solution with itself gives the energy bound

Let $H$ be a real or complex Hilbert space, $a$ a bounded coercive sesquilinear form with constant $\alpha >0$, $F$ a bounded conjugate-linear functional on $H$, and $u\in H$ a solution of $a(u,v)=F(v)$ for all $v\in H$. Then $\alpha \mathrm{\parallel }u{\mathrm{\parallel }}^2\le \mathrm{Re}a(u,u)=\mathrm{Re}F(u)\le \mathrm{\parallel }F\mathrm{\parallel }\mathrm{\parallel }u\mathrm{\parallel },\text{hence }\alpha \mathrm{\parallel }u\mathrm{\parallel }\le \mathrm{\parallel }F\mathrm{\parallel }\mathrm{.}$ The bound is a priori in the sense that it uses only the equation, coercivity and the norm of the datum, not the construction of $u$; it applies directly to homogeneous Dirichlet solutions $u\in H_0^1(\mathrm{\Omega })$ of Weak Dirichlet solutions for a divergence-form operator after substituting their coercivity constants. For an inhomogeneous Dirichlet solution, first subtract a lifting to obtain a solution in $H_0^1(\mathrm{\Omega })$ and use its residual datum; the original solution need not itself be an admissible test.