Best-approximation gap of a finite-dimensional eigenspace

Let $V$ be a real or complex Hilbert space with a bounded, Hermitian, coercive form $a$ as in Bounded, coercive and symmetric sesquilinear forms. Its energy norm is $\mathrm{\parallel }v{\mathrm{\parallel }}_a:=\sqrt{a(v,v)}$. Let $E\subseteq V$ be a nonzero finite-dimensional subspace and let $V_h\subseteq V$ be a finite-dimensional subspace. Define the directed energy distance and best-approximation gap by $d_a(v,V_h):={\inf }_{w_h\in V_h}\mathrm{\parallel }v-w_h{\mathrm{\parallel }}_a,{\delta }_a(E,V_h):={\sup }_{\begin{array}{c}v\in E\\ \mathrm{\parallel }v{\mathrm{\parallel }}_a=1\end{array}}{d}_a(v,V_h)\mathrm{.}$ When $E=\mathrm{span}\{e_1,\dots ,e_m\}$ is the first $m$ continuous eigenspace directions and $V_h$ is the conforming Galerkin space, write ${\delta }_h:={\delta }_a(E,V_h)$. This is a directed gap from $E$ to $V_h$; it does not by itself assert a two-sided gap between two eigenspaces.

The quantity is finite and satisfies $0\le {\delta }_a(E,V_h)\le 1$, since $0\in V_h$ gives $d_a(v,V_h)\le \mathrm{\parallel }v{\mathrm{\parallel }}_a$ on the unit sphere.