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In this paper we use a duality method to introduce a new space of generalized distributions. This method is exactly the same introduced by Schwartz for the distribution theory. Our space of generalized distributions contains all the Schwartz distributions and all the multipole series of physicists and is, in a certain sense, the smallest space containing all these series.

In this paper we define, by duality methods, a space of ultradistributions ${}_{\omega }^{\text{'}}\left({ℝ}^N\right)$. This space contains all tempered distributions and is closed under derivatives, complex translations and Fourier transform. Moreover, it contains some multipole series and all entire functions of order less than two. The method used to construct ${𝔾}_{\omega }^{\text{'}}\left({ℝ}^N\right)$ led us to a detailed study, presented at the beginning of the paper, of the duals of infinite dimensional locally convex spaces that are inductive limits of finite dimensional...