HOLOMORPHIC-FUNCTIONS AND BANACH-NUCLEAR DECOMPOSITIONS OF FRECHET SPACES

We introduce a decomposition of holomorphic functions on Frechet spaces which reduces to the Taylor series expansion in the case of Banach spaces and to the monomial expansion in the case of Frechet nuclear spaces with basis. We apply this decomposition to obtain examples of Frechet spaces E for which the tau(omega) and tau(delta) topologies on H(E) coincide. Our result includes, with simplified proofs, the main known results-Banach spaces with an unconditional basis and Frechet nuclear spaces with DN [2, 4, 5, 6]-together with new examples, e.g. Banach spaces with an unconditional finite-dimensional Schauder decomposition and certain Frechet-Schwartz spaces. This gives the first examples of Frechet spaces, which are not nuclear, with tau0 = tau(delta) on H(E).