Descriptions of Spaces Strongly Dual to Inductive Limits of Subspaces of \(\boldsymbol{H(D)}\)

In this paper, we consider the Banach space of analytic functions in a given bounded convex domain of a multidimensional complex space with a restriction on derivatives for each logarithmically convex sequence. On the union of such Banach spaces we define the linear topology of the canonical inductive limit and prove the completeness of the system of all complex exponentials. The main result is a description of the space of Laplace transforms of linear continuous functionals in the algebraic sense under certain conditions on sequences and their family. In the resulting weight space, we define the topology of the projective limit so that the Laplace transform establishes an isomorphism between the strongly conjugate space to the original space and the resulting projective limit of Banach spaces of entire functions. Previously, similar problems were considered for functions of a single variable. This work is the final part of the researches of I.Kh. Musin, R.S. Yulmukhametov, and the author for functions of many variables.