 These properties still do not quite suffice to capture \(L(\mathbf{H})\), but they do get us into the right ballpark. Let \(\mathbf{V}\) be any inner product space over an involutive division ring \(D\). A subspace \(\mathbf{M}\) of \(\mathbf{V}\) is said to be \(\bot\)-closed iff \(\mathbf{M} = \mathbf{M}^{\bot \bot}\), where \(\mathbf{M}^{\bot} = \{v\in \mathbf{V} \mid \forall m\in \mathbf{M}( \langle v,m\rangle = 0)\}\). Ordered by set-inclusion, the collection \(L(\mathbf{V})\) of all \(\bot\)-closed subspaces of \(\mathbf{V}\) forms a complete atomic lattice, orthocomplemented by the mapping \(\mathbf{M} \rightarrow \mathbf{M}^{\bot}\). A theorem of Amemiya and Araki (1966) shows that a real, complex or quaternionic inner product space \(\mathbf{V}\) with \(L(\mathbf{V})\) orthomodular, is necessarily complete. For this reason, an inner product space \(\mathbf{V}\) over an involutive division ring is called a generalized Hilbert space if its lattice\(L(\mathbf{V})\) of \(\bot\)-closed subspaces is orthomodular. The following representation theorem is due to C. Piron [1964]: