On some spaces of analytic functions and their duality relations

For each 0 ≤ C  < + ∞and 0 < p < +∞ let EC,p be the space of entire functions f such that, for some constant A  ≥ 0,|f(z) ≤  Ae^C|z|^p for all z in c. If ||f|| C, p is the minimun of such constants A, ||  ||c,p is a Banach space norm on EC,p.

Let 0 < B ≤ + ∞ and denote with EB, p the inductive limit space of the Banach spaces EC,p , 0 ≤  C < B. The topological dual space of EB, p is identified as the space 0B,p of analytic functions on the open disk D(0,(Bp)1/p). If 0B,P is given the topology of uniform convergence on compact sets, its topological dual is also identified as EB,p. Relations between different topologies on the spaces EC,p and EB, p having their origin in the duality are also examinea.