Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”: $$\operatorname{Vol}(L_V):=\sqrt{\det(B^\top B)},$$ where $B$ is any basis matrix of the lattice $L_V:=V\cap \Bbb Z^n$.
Question: Is it possible to extend this definition to non-rational $V$?
One idea I had is to approximate $V$ in the appropriate Grassmanian by rational subspaces. But one runs into the problem that the sequence of lattice volumes of the approximating subspaces might not converge, as can be seen from the sequence $V_j=\Bbb R (j, 1)$—this sequence converges to the $x$-axis, yet $\operatorname{Vol}(L_{V_j})= \sqrt{j^2+1}\to \infty$. In addition, I have no reason to think that given a sequence $V_j$ of rational subspaces converging to a rational subspace $V$ for which the sequence of lattice volumes does converge—say to a value $a$—that $a$ is necessarily $\operatorname{Vol}(L_V)$.