Ultradistributions of Roumieu Type and Projective Descriptions1

We show that the topology of the weighted inductive limit of Fr´echet spaces of entire functions which is obtained as the Fourier Laplace transform of the space of ultradistributions with compact support of Roumieu type cannot be described by means of canonical weighted sup-seminorms. This is a natural weighted LF-space of entire functions for which the problem of projective description of Bierstedt, Meise, and Summers has a negative answer. However, the projective hull of the LF-space coincides algebraically with the inductive limit. The behaviour of the corresponding inductive limits of spaces of continuous functions is completely opposite: the topology can be described by weighted sup-seminorms, but the LF-space is a proper subspace of its projective hull