$\DeclareMathOperator\Gr{Gr}$The Grassmnnian variety $\Gr(k,n)$ is the set of $k$-dimensional subspaces of $\mathbb{C}^n$. The coordinate ring $\mathbb{C}[\Gr(k,n)]$ is generated by Plucker coordinates subject to Plucker relations.

Denote by $\Gr(k, \pm \infty)$ the set of $k$-dimensional subspaces of $\mathbb{C}^{\infty}$. The coordinate ring $\mathbb{C}[\Gr(k,\pm \infty)]$ is generated by Plucker coordinates $P_J$, $J \subset \mathbb{Z}$, subject to Plucker relations. Has $\mathbb{C}[\Gr(k,\pm \infty)]$ been defined in the literature? Thank you very much.