Given a problem in weak formulation $$ \begin{align} \text{find $u\in V$ s.th. for all $v\in V$} \\ a(u,v) = f(v) \end{align} $$
with bilinear form $a:V\times V\rightarrow\mathbb{R}$, bounded with constant $k$ $$ a(u,v) \le k\|u\| \|v\|\quad \forall u,v\in V $$ and elliptic with constant $c$ $$ a(u,u) \ge c \|u\|^2\quad \forall u\in V $$ Denote by $u$ the solution to the weak problem in $V$ and by $u_m$ the solution to the same problem restricted to the $V_m$ of a Galerkin scheme for $V$, then Cea's lemma states that there holds a quasi-best approximation property $$ \|u-u_m\| \le \frac{k}{c} \inf_{v\in V_m} \|u-v\| \,. $$ For symmetric $a(u, v) = a(v, u)$ the coefficient can be improved to $\sqrt{\frac{k}{c}}$.
My question is now the following: Is this bound sharp? Ie. are there examples where equality holds? If not, are there any lower bounds known?
Look at the proof. If there is equality in the hypothesis ($a(x,y) = c\times x\cdot y$) then there is equality also in the Cea's lemma.