The property (DN) and the exponential representation of holomorphic functions

Abstract

It is shown that a nuclear Frechet spaceE has the property (DN) if and only if every holomorphic function onE *, the strongly dual space ofE, with values in the strongly dual space of a Frechet spaceF having the property ( $$\bar \Omega$$ ) can be represented in the exponential form. Moreover, it is shown that the space of holomorphic functions onC ∞, the space of all complex number sequences, has a linearly absolutely exponential representation system. But the space of holomorphic functions onE * does not have such a system whenE is a nuclear Frechet space that does not have the property (DN).