On kernel theorems for (LF)-spaces

A convenient technique for proving kernel theorems for (LF)-spaces (countable inductive limits of Frechet spaces)is developed. The proposed approach is based on introducing a suitable modification of the functor of the completed inductive topological tensor product. Using such modified tensor products makes it possible to prove kernel theorems without assuming the completeness of the considered (LF)-spaces. The general construction is applied to proving kernel theorems for a class of spaces of entire analytic functions arising in nonlocal quantum field theory.