Holomorphic functions of uniformly bounded type and linear topological invariants \((H_{ub})\), \((\widetilde{LB}^{\infty})\) and \((\widetilde{LB}_{\infty})\) (Q1430483)

Given Fréchet spaces \(E\) and \(F\), \({\mathcal H}(E,F)\) denotes the space of all holomorphic functions from \(E\) into \(F\). Certain subspaces of \({\mathcal H}(E,F)\) arise naturally. Among these are \({\mathcal H}_b(E,F)\), the space of all \(f\) in \({\mathcal H}(E,F)\) which are bounded on bounded subsets of \(E\), and \({\mathcal H}_{ub}(E,F)\), the space of all \(f\) in \({\mathcal H}(E,F)\) which are bounded on all multiples of some neighbourhood of \(0\). In general, \({\mathcal H}_{ub}(E,F)\subseteq {\mathcal H}_b(E,F)\subseteq {\mathcal H}(E,F)\). A locally convex space \(E\) is said to belong to \((H_{ub})\) if \({\mathcal H}(E,\mathbb{C})={\mathcal H}_{ub}(E,\mathbb{C})\). This paper is concerned with determining conditions on \(E\) and \(F\) that will ensure \({\mathcal H}_b(E,F)={\mathcal H}_{ub}(E,F)\). The author introduces two new linear topological invariants, \((\widetilde{LB^\infty})\) and \((\widetilde{LB_\infty})\), on Fréchet spaces. These invariants are in the spirit of the invariants of \textit{D. Vogt} [J. Reine Angew. Math. 345, 182--200 (1983; Zbl 0514.46003), Manuscr. Math. 37, 269--301 (1982; Zbl 0512.46003)]. The invariant \((\widetilde{LB^\infty})\), is shown to be strictly weaker than the invariant \((\widetilde\Omega)\), while every space with \((\overline{DN})\) has \((\widetilde{LB_\infty})\). When \(E\) is a Fréchet space with an absolute basis, the author proves that \({\mathcal H}_b(E,F)={\mathcal H}_{ub}(E,F)\) if (i) \(E\) has both \(({\mathcal H}_{ub})\) and \((\widetilde{LB^\infty})\) and \(F\) has \((LB_\infty)\), or (ii) \(E\) has \(({\mathcal H}_{ub})\) and \(F\) has \((\widetilde{LB_\infty})\). (For the definitions of \((\widetilde\Omega)\), \((\overline{DN})\) and \((LB_\infty)\), see \textit{D. Vogt}, loc. cit.)