  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.
