(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2005)
We consider a complex submanifold X of finite codimension in an infinite-dimensional complex projective space P and prove that the first Dolbeault cohomology group of all line bundles O_X(n), n Z, vanishes when X is a complete intersection and P admits smooth partitions of unity.