We introduce a normed space of functions, holomorphic in a bounded convex domain. Its elements are infinitely differentiable up to the boundary, and all their derivatives satisfy estimates specified by a convex sequence of positive numbers. We consider its largest linear subspace that is invariant with respect to the operator of differentiation and provide it with the natural topology of projective limit. We establish duality between this subspace and some space of entire functions. Based on this, we construct a representing system of exponentials in the subspace.