Holomorphic functions on strict inductive limits

We study strict inductive limits of Fréchet Montel (FM) spaces and reflexive Fréchet (RF) spaces and we obtain some interesting examples in the theory of infinite dimensional holomorphy. P_M(^kE′) and P_HY(^kE′) will denote respectively the set of all k-homogeneous polynomials on E′ that are bounded on bounded sets and the set of all k-homogeneous polynomials on E′ that are continuous on compact sets. ?_SM(^kE′) is the space of all symetric k -multilinear mappings from E′ × ... × E′ into C that are bounded on bounded sets. H_HY(E′) will denote the set of all G-analytic functions on E′ that are continuous on the compact subsets of E′.