Union of linear spaces in infinite-dimensional projective spaces and holomorphic vector bundles

Let V be a localizing Banach space with an unconditional countable basis, X an equicodimensional transversal union of finite-codimensional linear subspaces of P(V) and E a holomorphic vector bundle of finite rank on X. Here we prove that Hⁱ(X,E) = 0 for every i > 0 and that E is isomorphic to a direct sum of line bundles ^OX(t).