  The Barth-Van de Ven-Tyurin-Sato Theorem claims that any finite rank vector
bundle on the infinite complex projective space $\mathbf{P}^\infty$ is
isomorphic to a direct sum of line bundles. We establish sufficient conditions
on a locally complete linear ind-variety $\mathbf{X}$ which ensure that the
same result holds on $\mathbf{X}$. We then exhibit natural classes of locally
complete linear ind-varieties which satisfy these sufficient conditions.
  Keywords: ind-variety, vector bundle
