\(\alpha\) \(\mu\)-duals and holomorphic (nuclear) mappings (Q1085390)

Corresponding to a sequence space (s.s) \(\mu\) and a sequence \(\alpha\), we introduce the notion of an \(\alpha\mu\)-dual \(\lambda^{\mu}_{\alpha}\) of a s.s. \(\lambda\) as follows: \[ \lambda^{\mu}_{\alpha}=\{b\in \omega: \alpha ab\in \mu,\quad \forall a\in \lambda \}. \] This envelops as particular cases, the concepts of Köthe, \(\beta\)-, \(\gamma\)-duals and various other spaces, for instance the space introduced by \textit{P. J. Boland} in J. Reine Angew. Math. 270, 38-60 (1974; Zbl 0323.46042). We prove some results generalizing the Köthe theory of s.s. for \(\alpha\mu\)-duals and give several examples. Using these concepts, we consider several subspaces of the class of holomorphic mappings defined on a Banach space and study their structural properties including characterization of bounded and compact subsets in these subspaces. Finally, we show that the class of monomials forms a fully \(\ell^ 1\)- base for the class of hypoanalytic functions defined on a normal open subset of the strong Köthe dual of a barrelled nuclear s.s.