Suppose E is a separated
complex locally convex space, U is non void open subset of E, F a complex
normed space and ℋ(U;F) the complex vector space of all holomorphic
mappings from U into F. On ℋ(U;F) we consider the following topologies; a)
τ_{ωs}, the topology generated by the seminorms p which are K − B ported
for some K ⊂ U compact and B ⊂ E bounded. A seminorm p is K − B
ported if for every 𝜖 > 0, with K + 𝜖B ⊂ U, there is c(𝜖) > 0, such that
p(f) ≦ c(𝜖)sup{∥f(x)∥;x ∈ K + 𝜖B} for all f ∈ℋ(U;F); b) τ_{0}, the compact open
topology; c) τ_{∞s} the topology defined by J. A. Barroso in “Topologias nos
espaços de aplicações holomorfas entre espaços localmente convexos”,
An. Acad. Brasil. Ci, 43, 1971. The topology τ_{ωs} is an generalization of the
Nachbin topology (L. Nachbin, Topology on Spaces of Holomorphic Mappings,
SpringerVerlag, 1968). The following results are valid: 1. ℋ⊂ℋ(U;F) is
τ_{0}bounded if, and only if, ℋ is τ_{ωs}bounded. 2. ℋ⊂ℋ(U;F) is τ_{ωs}relatively
compact if, and only if, ℋ is τ_{∞s}relatively compact. 3. Let E be a quasi
complete space. Then τ_{0} = τ_{ωs} on ℋ(E;C) if, and only if E is a semiMontel
space. Moreover, the completion of ℋ(E;C) on the τ_{ωs} topology and the
bornological topology associated to τ_{0} are characterized via the Silvaholomorphic
mappings.
