Definition: Representability on a declared space

A real linear functional $B$ on $\mathcal{V}\subseteq \mathcal{S}$ is FH-representable on $\mathcal{V}$ if a single $J\in \mathcal{E}$ satisfies $B(h)=F_J(h)$ for every $h\in \mathcal{V}$. For the non-spacelike-compact TT plane below, the same compact pairing defines a restricted extension, without identifying that plane with the phase space of an FH algebra.