Let$U$be an open subset of a complex locally convex space$E$, and$H(U)$the space of holomorphic functions from$U$to$C$. If the dual$E′$of$E$is nuclear with respect to the topology generated by the absolutely convex compact subsets of$E$, then it is shown that$H(U)$endowed with the compact open topology is a nuclear space. In particular, if$E$is the strong dual of a Fréchet nuclear space, then$H(U)$is a Fréchet nuclear space.