  We consider ind-varieties obtained as direct limits of chains of embeddings
$X_1\stackrel{\phi_1}{\hookrightarrow}\dots\stackrel{\phi_{m-1}}{\hookrightarrow}
X_m\stackrel{\phi_m}{\hookrightarrow}X_{m+1}\stackrel{\phi_{m+1}}{\hookrightarrow}\dots$,
where each $X_m$ is a Grassmannian or an isotropic Grassmannian (possibly
mixing Grassmannians and isotropic Grassmannians), and the embeddings $\phi_m$
are linear in the sense that they induce isomorphisms of Picard groups. We
prove that any such ind-variety is isomorphic to one of certain standard
ind-Grassmannians and that the latter are pairwise non-isomorphic
ind-varieties.
