The elliptic Ritz projection for a conforming finite-element space

Let $(V,H,V^{\ast })$ be a real evolution triple and let $V_h\subset V$ be a finite-dimensional conforming subspace. In a mesh instance, the algebraic piecewise-polynomial conventions are those of Broken and conforming polynomial spaces on a matching mesh; the required inclusion into $V$ is an explicit hypothesis of this definition. Let $a:V\times V\to \mathbb{R}$ be time-independent, bounded, and coercive as in Bounded, coercive and symmetric sesquilinear forms, with constants $M$ and $\alpha >0$. For $u\in V$, the equation for the unknown $z_h\in V_h$ $a(z_h,v_h)=a(u,v_h)\text{for every }{v}_h\in V_h$ is the Ritz equation; any solution $z_h$ is called a Ritz projection of $u$. The item Existence, uniqueness, linearity, and boundedness of the Ritz projection ↗ proves that for every $u$ there is exactly one solution, denoted $R_{h}u$, and that these solutions define a bounded linear map $R_h:V\to V_h$. The form need not be symmetric.