  The simplest example of an ind-Grassmannian is the infinite projective space
$\mathbf P^\infty$. The Barth-Van de Ven-Tyurin (BVT) Theorem, proved more than
30 years ago \cite{BV}, \cite{T}, \cite{Sa} (see also a recent proof by A.
Coand\u{a} and G. Trautmann, \cite{CT}), claims that any vector bundle of
finite rank on $\mathbf P^\infty$ is isomorphic to a direct sum of line
bundles. In the last decade natural examples of infinite flag varieties (or
flag ind-varieties) have arisen as homogeneous spaces of locally linear
ind-groups, \cite{DPW}, \cite{DiP}. In the present paper we concentrate our
attention to the special case of ind-Grassmannians, i.e. to inductive limits of
Grassmannians of growing dimension.
